Analysis of a splitting method for stochastic balance laws

Analysis of a splitting method for stochastic balance laws Abstract We analyse a semidiscrete splitting method for conservation laws driven by a semilinear noise term. Making use of fractional bounded variation (BV) estimates, we show that the splitting method generates approximate solutions converging to the exact solution, as the time step $$\Delta t \to 0$$. Under the assumption of a homogenous noise function, and thus the availability of BV estimates, we provide an $$L^1$$-error estimate. Bringing into play a generalization of Kružkov’s entropy condition, permitting the ‘Kružkov constants’ to be Malliavin differentiable random variables, we establish an $$L^1$$-convergence rate of order $$\frac13$$ in $$\Delta t$$. 1. Introduction Recently, there have been many works studying the effect of stochastic forcing on scalar conservation laws (Vallet, 2000; Kim, 2003; Feng & Nualart, 2008; Vallet & Wittbold, 2009; Debussche & Vovelle, 2010, 2015; Bauzet et al., 2012; Chen et al., 2012; Hofmanov´a, 2013; Biswas et al., 2015; Karlsen & Storrøsten, 2017; Debussche & Vovelle, 2015; Debussche et al., 2016) with emphasis on existence, uniqueness and stability questions. Deterministic conservation laws exhibit shocks (discontinuous solutions), and a weak formulation coupled with an appropriate entropy condition is required to ensure the well-posedness (Kružkov, 1970). The question of uniqueness gets somewhat more difficult by adding a stochastic source term due to the interaction between noise and nonlinearity. A pathwise theory for conservation laws with stochastic fluxes has been developed in Lions et al. (2013, 2014) and Gess & Souganidis (2014, 2015). In this article, we are interested in the convergence of approximate solutions to conservation laws driven by a multiplicative Wiener noise term, i.e., stochastic balance laws of the form   du+divf(u)dt=σ(x,u)dB,(t,x)∈ΠT (1.1) with initial data   u(0,x)=u0(x),x∈Rd. (1.2) We denote by $$\nabla$$ and $${\operatorname{div}}=\nabla\cdot$$ the spatial gradient and divergence, respectively. Moreover, $${\it {\Pi}}_T = {\mathbb{R}}^d \times (0,T)$$ for some fixed final time $$T>0$$, and $$u(x,t)$$ is the scalar unknown function that is sought. The random force in (1.1) is driven by a Wiener process $$B=B(t)=B(t,\omega)$$, $$\omega\in {\it {\Omega}}$$, over a stochastic basis $$({\it {\Omega}},{\mathscr{F}},\left\{{{\mathscr{F}}_t}\right\}_{t\ge 0},P)$$, where $$P$$ is a probability measure, $${\mathscr{F}}$$ is a $$\sigma$$-algebra and $$\left\{{{\mathscr{F}}_t}\right\}_{t\ge 0}$$ is a right-continuous filtration on $$({\it {\Omega}},{\mathscr{F}})$$ such that $${\mathscr{F}}_0$$ contains all the $$P$$-negligible subsets. The convection flux $$f:{\mathbb{R}}\to{\mathbb{R}}^d$$ satisfies   f is (globally) Lipschitz continuous on R. (𝓐f) Furthermore, we will sometimes make use of the assumption   f″ is uniformly bounded on R. (𝓐f,1) The noise coefficient $$\sigma:{\mathbb{R}}^d\times {\mathbb{R}}\to {\mathbb{R}}$$ is assumed to satisfy   ‖σ‖Lip=supx∈Rdsupu≠v{|σ(x,u)−σ(x,v)||u−v|}<∞,|σ(⋅,0)|∈L∞(Rd). (𝓐σ) These assumptions imply   |σ(x,u)−σ(x,v)|≤‖σ‖Lip|u−v|,|σ(x,u)|≤max{‖σ‖Lip,‖σ(⋅,0)‖L∞(Rd)}(1+|u|). Furthermore, we often assume the existence of constants $$M_\sigma$$ and $${\kappa_\sigma}$$ such that   |σ(x,u)−σ(y,u)|≤Mσ|x−y|κσ+1/2(1+|u|),κσ∈(0,1/2]. (𝓐σ,1) A prevailing difficulty affecting convergence/error analysis is related to the time discretization and the interplay between noise and nonlinearity. Up to now, there have been only a few studies investigating this problem. Holden & Risebro (1997) study a one-dimensional equation with bounded initial data and a compactly supported, homogeneous noise function $$\sigma=\sigma(u)$$, ensuring $$L^\infty$$-bounds on the solution. An operator splitting method is used to construct approximate solutions, and it is shown that a subsequence of these approximations converges to a (possibly nonunique) weak solution. Recently, this work was generalized to stochastic entropy solutions and extended to the multidimensional case by Bauzet (2015). Kröker & Rohde (2012) analyse semidiscrete (time-continuous) finite volume methods. They use the compensated compactness method to prove convergence to a stochastic entropy solution for one-dimensional equations, with nonhomogeneous noise function $$\sigma=\sigma(x,u)$$. Bauzet et al. (2016) analyse fully discrete finite volume methods for multidimensional equations, with homogeneous noise function $$\sigma=\sigma(u)$$. Their proof relies on weak bounded variation (BV) (energy) estimates and a uniqueness result for measure-valued stochastic entropy solutions. In this article (as in Holden & Risebro, 1997; Bauzet, 2015), we will investigate the semidiscrete splitting method for calculating approximations to stochastic entropy solutions of (1.1). Generally, this method is based on ‘splitting off’ the effect of the stochastic source $$\sigma(x,u)\, \mathrm{d} B$$. This Godunov-type operator splitting can be used to extend sophisticated numerical methods for deterministic conservation laws to stochastic balance laws. Generally, the tag ‘operator splitting’ refers to the well-known idea of constructing numerical methods for complicated partial differential equations by reducing them to a progression of simpler equations, each of which can be solved by some tailor-made numerical method. The operator splitting approach is described in a large number of articles and books. We do not survey the literature here, referring the reader instead to the bibliography in Holden et al. (2010). The main focus of the book (Holden et al., 2010) is on convergence results, within classes of discontinuous functions, for general splitting algorithms for deterministic nonlinear partial differential equations. Compared with the earlier results of Holden & Risebro (1997) and Bauzet (2015), the main contributions of this article are twofold. First, we establish convergence of the splitting approximations to a stochastic entropy solution in the case of nonhomogeneous noise functions $$\sigma = \sigma(x,u)$$. Whenever $$\sigma$$ has a dependency on the spatial position $$x$$, BV estimates are no longer available and the approach resorted to in Holden & Risebro (1997) and Bauzet (2015) does not apply. Following an idea laid out in Chen et al. (2012), and independently in Debussche & Vovelle (2010), we derive a fractional $$BV_x$$ estimate, which, via an interpolation argument à la Kružkov, is turned into a temporal equicontinuity estimate. These a priori estimates, along with Young measures and an earlier uniqueness result, are used to show that splitting approximations converge to a stochastic entropy solution. Let us make a few comments about the convergence proof. In the deterministic case, the spatial and temporal estimates would imply strong ($$L^1$$) compactness of the splitting approximations. In the stochastic setting, we have the randomness variable $$\omega$$ for which there is no compactness; as a matter of fact, possible ‘oscillations’ in $$\omega$$ may prevent strong compactness. In the literature, the standard way of dealing with this issue is to look for tightness (weak compactness) of the probability laws of the approximations. Then an application of the Skorokhod representation theorem provides a new probability space and new random variables, with the same laws as the original variables, that do converge strongly (almost surely) in $$\omega$$ to some limit. Equipped with almost sure convergence, it is not difficult to show that the limit variable is a so-called martingale solution, i.e., the limit is probabilistic weak in the sense that the stochastic basis is now viewed as part of the solution. One can pass (à la Yamada & Watanabe) from martingale to pathwise solutions provided there is a strong uniqueness result. In this article, we will not follow this ‘traditional’ approach. Instead, we will utilize Young measures, parametrized over $$(t,x,\omega)$$, to represent weak limits of nonlinear functions, thereby obtaining weak convergence of the splitting approximations toward a so-called Young measure-valued stochastic entropy solution. We use the spatial and temporal translation estimates to conclude that the limit is a solution in this sense. Weak convergence is then upgraded to strong convergence in $$(t,x,\omega)$$a posteriori, thanks to the fact that these measure-valued solutions are $$L^1$$-stable (unique). After the works of Tartar, DiPerna and others, weak compactness arguments of this type (propagation of compactness) are frequently used in the nonlinear partial differential equations (PDE) literature (cf., e.g., Szepessy, 1989; Málek et al., 1996; Panov, 1996; Eymard et al., 2000) and recently in the context of stochastic equations (Vallet & Wittbold, 2009; Bauzet et al., 2012, 2016; Bauzet, 2015; Biswas et al., 2015; Karlsen & Storrøsten, 2017). Our second main contribution is an $$L^1$$-error estimate of the form $${\mathcal O}(\Delta t^{1/3})$$, for homogeneous noise functions $$\sigma=\sigma(u)$$. Except for the expected convergence rate for the vanishing viscosity method (Chen et al., 2012), this appears to be the first error estimate derived for approximate solutions to stochastic conservation laws. The rate $$\frac13$$ should be compared with the first-order convergence rate available for conservation laws with deterministic source (Langseth et al., 1996). Our proof relies on BV estimates and a generalization of the Kružkov entropy condition, allowing the ‘Kružkov constants’ to be Malliavin differentiable random variables, which was put forward in the recent work Karlsen & Storrøsten (2017). The remaining part of this article is organized as follows: Section 2 collects some preliminarily material along with the relevant notion of (stochastic entropy) solution. The operator splitting method is defined precisely in Section 3. A series of a priori estimates are derived in Section 4, which are subsequently used in Section 5 to prove convergence toward a stochastic entropy solution. Section 6 is devoted to the proof of the error estimate. Section A is an appendix collecting some definitions and results used elsewhere in the paper. 2. Preliminaries In this article, as in Karlsen & Storrøsten (2017), we apply certain weighted $$L^p$$ spaces. Since we do not assume $$\sigma(x,0) \equiv 0$$, weighted spaces on $${\mathbb{R}}^d$$ provide a convenient alternative to working on the torus as in Debussche & Vovelle (2010) and Debussche et al. (2016). The weights used herein turn out to be suitable also for fractional $$BV_x$$ estimates; cf. Proposition 4.4. Let $$\mathfrak{N}$$ be the set of all nonzero $$\phi \in C^1({\mathbb{R}}^d) \cap L^1({\mathbb{R}}^d)$$ for which there exists a constant $$C$$ such that $$\left|{\nabla\phi}\right| \le C \phi$$. An example is $$\phi(x) = \exp^{-\sqrt{1 + \left|{x}\right|^2}}$$. Set   Cϕ=inf{C∣||∇ϕ|≤Cϕ}. For $$\phi \in \mathfrak{N}$$, we use the weighted $$L^p$$-norm $$\left\Vert {\cdot}\right\Vert_{p,\phi}$$ defined by   ‖u‖p,ϕ:=(∫Rd|u(x)|pϕ(x)dx)1/p. The corresponding weighted $$L^p$$-space is denoted by $$L^p({\mathbb{R}}^d,\phi)$$. Similarly, we define   XXX‖u‖∞,ϕ−1:=supx∈Rd{|u(x)|ϕ(x)},u∈C(Rd). (2.1) Some useful results regarding functions in $$\mathfrak{N}$$ are collected in Section A.2. We denote by $$\mathscr{E}$$ the set of non-negative convex functions in $$C^2({\mathbb{R}})$$ such that $$S'$$ is bounded and $$S''$$ compactly supported. A pair of functions $$(S,Q) $$ is called an entropy/entropy–flux pair if $$S:{\mathbb{R}}\to{\mathbb{R}}$$ is $$C^2$$ and $$Q=(Q_1,\ldots,Q_d):{\mathbb{R}} \mapsto{\mathbb{R}}^d $$ satisfies $$Q' = S' f'$$. An entropy/entropy–flux pair $$(S,Q)$$ is said to belong to $$\mathscr{E}$$ if $$S$$ belongs to $$\mathscr{E}$$. Let $${\mathscr{P}}$$ denote the predictable $$\sigma$$-algebra on $$[0,T] \times {\it {\Omega}}$$ with respect to $$\left\{{{\mathscr{F}}_t}\right\}$$ (see, e.g., (see, e.g., Chung & Williams, 2014, Section 2.2). In general we are working with equivalence classes of functions with respect to the measure $$\mathrm{d}t \otimes {\rm d}P$$. The equivalence class $$u$$ is said to be predictable if it has a version $$\tilde{u}$$ that is $${\mathscr{P}}$$-measurable. Equivalently, we could ask for any representative to be $${\mathscr{P}}^*$$-measurable, where $${\mathscr{P}}^*$$ is the completion of $${\mathscr{P}}$$ with respect to $$\mathrm{d}t \otimes {\rm d}P$$. Note that any (jointly) measurable and adapted process is $${\mathscr{P}}^*$$-measurable (cf., e.g., Chung & Williams, 2014, Theorem 3.7). Next we collect some basic material related to Malliavin calculus. We refer to Nualart (2006) for an introduction to the topic. The Malliavin calculus is developed with respect to the isonormal Gaussian process $$W:L^2([0,T]) \rightarrow \mathcal{H}^1$$, defined by $$W(h) := \int_0^Th\,\mathrm{d}B$$. Here $$\mathcal{H}^1$$ is the subspace of $$L^2({\it {\Omega}},{\mathscr{F}},P)$$ consisting of zero-mean Gaussian random variables. We denote by $$\mathcal{S}$$ the class of smooth random variables of the form   V=f(W(h1),…,W(hn)), where $$f \in C^\infty_c({\mathbb{R}}^n)$$, $$h_1, \dots, h_n \in L^2([0,T])$$ and $$n \geq 1$$. For such random variables, the Malliavin derivative is defined by   DV=∑i=1n∂if(W(h1),…,W(hn))hi, where $$\partial_i$$ denotes the derivative with respect to the $$i$$th variable. The space $$\mathcal{S}$$ is dense in $$L^2({\it {\Omega}},{\mathscr{F}},P)$$. Furthermore, the operator $$D$$ is closable as a map from $$L^2({\it {\Omega}})$$ to $$L^2({\it {\Omega}};L^2([0,T]))$$ (Nualart, 2006, Proposition 1.2.1). The domain of $$D$$ in $$L^2({\it {\Omega}})$$ is denoted by $${\mathbb{D}}^{1,2}$$. That is, $${\mathbb{D}}^{1,2}$$ is the closure of $$\mathcal{S}$$ with respect to the norm   ‖V‖D1,2={E[|V|2]+E[‖DV‖L2([0,T])2]}1/2. For the generalization of the above to Hilbert space-valued random variables (see Nualart, 2006, Remark 2, p. 31). We use the notion of stochastic entropy solution introduced in Karlsen & Storrøsten (2017), which is a refinement of the notion introduced by Feng & Nualart (2008). Definition 2.1 Fix $$\phi \in \mathfrak{N}$$. A stochastic entropy solution $$u$$ of (1.1) and (1.2) with $$u_0 \in L^2({\it {\Omega}},{\mathscr{F}}_0,P;L^2({\mathbb{R}}^d,\phi))$$ is a stochastic process   u={u(t)=u(t,x)=u(t,x;ω)}t∈[0,T] satisfying the following conditions: (i) $$u$$ is a predictable process in $$L^2([0,T] \times {\it {\Omega}};L^2({\mathbb{R}}^d,\phi))$$. (ii) For any random variable $$V \in {\mathbb{D}}^{1,2}$$ and any entropy, entropy–flux pair $$(S,Q) \in \mathscr{E}$$,   E[∬ΠTS(u−V)∂tφ+Q(u,V)⋅∇φdxdt+∫RdS(u0(x)−V)φ(0,x)dx] −E[∬ΠTS″(u−V)σ(x,u)DtVφdxdt] +12E[∬ΠTS″(u−V)σ(x,u)2φdxdt]≥0 for all non-negative $$\varphi \in C^\infty_c([0,T) \times {\mathbb{R}}^d)$$. Here, $$L^2([0,T] \times {\it {\Omega}};L^2({\mathbb{R}}^d,\phi))$$ denotes the Lebesgue–Bochner space and $$D_tV$$ denotes the Malliavin derivative of $$V$$ evaluated at time $$t$$. By Karlsen & Storrøsten (2017, Lemma 2.2), it suffices to consider $$V \in \mathcal{S}$$ in (ii). In Karlsen & Storrøsten (2017), the existence and uniqueness of entropy solutions in the sense of Definition 2.1 is established under assumptions (𝓐f), (𝓐σ) and (𝓐σ,1). We also mention that whenever $$u_0 \in L^p({\it {\Omega}};L^p({\mathbb{R}}^d,\phi))$$ with $$2 \le p < \infty$$,   ess sup0≤t≤T{E[‖u(t)‖p,ϕp]}<∞. Let $$\left\{{J_\delta}\right\}_{\delta > 0}$$ be a sequence of symmetric mollifiers on $${\mathbb{R}}^d$$, i.e.,   Jδ(x)=1δdJ(xδ), (2.2) where $$J \geq 0$$ is a smooth, symmetric function satisfying $$\text{supp}\, (J)\subset B(0,1)$$ and $$\int J=1$$. For $$d = 1$$, we set $$J^+(x) = J(x - 1)$$, so that $$\text{supp}\,(J^+) \subset (0,2)$$. Under the additional assumption (𝓐σ,1), Karlsen & Storrøsten (2017, Proposition 5.2) assert that the entropy solution $$u$$ satisfies   E[∬Rd×Rd|u(t,x+z)−u(t,x−z)|Jr(z)ϕ(x)dx] ≤eCϕ‖f‖LiptE[∬Rd×Rd|u0(x+z)−u0(x−z)|Jr(z)ϕ(x)dx]+O(rκσ), (2.3) where $${{\kappa_\sigma}}$$ is given in (𝓐σ,1). Whenever $$\sigma(x,u) = \sigma(u)$$, the last term on the right-hand side vanishes, i.e., $${\mathcal{O}}(\ldots) = 0$$. 3. Operator splitting We will now describe the basic operator splitting method for (1.1). Let $$\mathcal{S}_{\text{CL}}(t)$$ be the solution operator that maps an initial function $$v_0(x)$$ to the unique entropy solution of the deterministic conservation law   ∂tv+divf(v)=0,v(0,x)=v0(x), (3.1) i.e., if $$v(t) := \mathcal{S}_{\text{CL}}(t)v_0$$ then $$v$$ is the unique entropy solution of (3.1). More precisely, for each $$\tau \in [0,T]$$,    ∫Rd|v0(x)−c|φ(0,x)dx−∫Rd|v(τ)−c|φ(τ,x)dx +∫0τ∫Rd|v−c|∂tφ+sign(v−c)(f(v)−f(c))⋅∇φdxdt≥0 for all $$c \in {\mathbb{R}}$$ and all non-negative $$\varphi \in C^\infty_c([0,T) \times {\mathbb{R}})$$. Note that the integrals are well defined due to the global Lipschitz assumption (𝓐f). Recall that the entropy solution has a version that belongs to $$C([0,T];L^1_{\mathrm{loc}}({\mathbb{R}}^d))$$ (Cancès & Gallouët, 2011). As we frequently need to consider the evaluation $$v(t),$$ it is convenient for us to assume that $$v$$ has this property. Let $$u,v \in L^1({\mathbb{R}}^d,\phi),$$ where $$\phi \in \mathfrak{N}$$. Then, for any $$t \in [0,T]$$,   ‖SCL(t)v−SCL(t)u‖1,ϕ≤eCϕ‖f‖Lipt‖u−v‖1,ϕ. Suppose $$u \in L^1({\it {\Omega}},{\mathscr{F}}_s,P;L^1({\mathbb{R}}^d,\phi))$$ for some $$s \in [0,T]$$. Let $$s \leq t \leq T$$. By considering the composition $${\it {\Omega}} \ni \omega \mapsto u(\omega) \mapsto \mathcal{S}_{\text{CL}}(t-s)u(\omega)$$, it follows that $$\mathcal{S}_{\text{CL}}(t-s)u$$ is $${\mathscr{F}}_s$$-measurable as an element in $$L^1({\mathbb{R}}^d,\phi)$$ (cf. Mishra & Schwab, 2012, Section 3.3). Similarly, for $$s \leq t \leq T$$, we let $${\mathcal{S}_{\text{SDE}}}(t,s)$$ denote the two-paramater semigroup defined by $${\mathcal{S}_{\text{SDE}}}(t,s)w^s = w(t)$$, where $$w$$ is the strong solution of   w(t,x)=ws(x)+∫stσ(x,w(r,x))dB(r). Suppose $$w^s,v^s \in L^1({\it {\Omega}},{\mathscr{F}}_s,P;L^1({\mathbb{R}}^d,\phi))$$. Then,   E[‖SSDE(t,s)ws−SSDE(t,s)vs‖1,ϕ]=E[‖ws−vs1,ϕ‖]. (3.2) To see this, let $$S_\delta \rightarrow \left|{\cdot}\right|$$ as $$\delta \downarrow 0$$ and consider the quantity $$S_\delta(w(t,x)-v(t,x))$$. Next, apply Itô’s formula, multiply by $$\phi$$ and let $$\delta \downarrow 0$$. Because of (3.2),   SSDE(⋅,s):L1(Ω,Fs,P;L1(Rd,ϕ))→L1([s,T]×Ω,P[s,T],dt⊗dP;L1(Rd,ϕ)), where $${\mathscr{P}}_{[s,T]}$$ denotes the predictable $$\sigma$$-algebra relative to $$\left\{{{\mathscr{F}}_t}\right\}_{s \le t \le T}$$ on $$[s,T] \times {\it {\Omega}}$$. Fix $$N \in {\mathbb{N}}$$, specify $${{{\it {\Delta}} t}} = T/N$$ and set $$t_n = n{{{\it {\Delta}} t}}$$. Let $$u^0 = u^0(x;\omega)$$ be given. The operator splitting, with initial condition $$u^0$$, is the sequence $$\left\{{u^n = u^n(x;\omega)}\right\}_{n = 0}^N$$ defined recursively by   un+1(x;ω)=[SSDE(tn+1,tn;ω)∘SCL(Δt)]un(x;ω) (3.3) for $$n = 0,1, \dots ,N-1$$. A graphical representation is given in Fig. 1. Fig. 1. View largeDownload slide A graphical representation of $$\left\{{u^n}\right\}$$, $${u_{{{{\it {\Delta}} t}}}}$$ and $${v_{{{{\it {\Delta}} t}}}}$$. Fig. 1. View largeDownload slide A graphical representation of $$\left\{{u^n}\right\}$$, $${u_{{{{\it {\Delta}} t}}}}$$ and $${v_{{{{\it {\Delta}} t}}}}$$. To investigate the convergence of the semidiscrete splitting algorithm (3.3), we need to work with functions that are defined not only for each $$t_n = n{{{\it {\Delta}} t}}$$ but also for the entire interval $$[0,T]$$. To this end, we introduce two different ‘time interpolants’ $${u_{{{{\it {\Delta}} t}}}}(t)={u_{{{{\it {\Delta}} t}}}}(t,x;\omega)$$ and $${v_{{{{\it {\Delta}} t}}}}(t)={v_{{{{\it {\Delta}} t}}}}(t,x;\omega)$$, defined for $$n=0,\ldots,N-1$$ by   uΔt(t)=SSDE(t,tn)∘SCL(Δt)un,t∈(tn,tn+1] (3.4) and   vΔt(t)=SCL(t−tn)un,t∈[tn,tn+1), (3.5) respectively; cf. Fig. 1. As $${u_{{{{\it {\Delta}} t}}}}$$ is discontinuous at $$t_n,$$ we introduce the right limit $${u_{{{{\it {\Delta}} t}}}}((t_n)+) = \mathcal{S}_{\text{CL}}({{{\it {\Delta}} t}})u^n$$. Similarly, let $${v_{{{{\it {\Delta}} t}}}}((t_{n+1})-) = \mathcal{S}_{\text{CL}}({{{\it {\Delta}} t}})u^n$$. 4. A priori estimates To establish the convergence of $$\left\{{{u_{{{{\it {\Delta}} t}}}}}\right\}_{{{{\it {\Delta}} t}}>0}, \left\{{{v_{{{{\it {\Delta}} t}}}}}\right\}_{{{{\it {\Delta}} t}} > 0}$$ we will need a series of a priori estimates. These are also crucial when deriving the error estimate. The following result explains the introduction of the weight functions $$\mathfrak{N}$$. Proposition 4.1 (Local $$L^p$$ estimates). Suppose (𝓐f) and (𝓐σ) are satisfied, $$2 \le p < \infty$$ and $$M \ge \left\Vert{f}\right\Vert_{\mathrm{Lip}}$$. Let $$\left\{{u^n}\right\}$$ be the splitting solutions defined by $$(3.3)$$, with initial condition $$u^0 \in L^p({\it {\Omega}},{\mathscr{F}}_0,P;L^p_\mathrm{loc}({\mathbb{R}}^d))$$. For $$t\in (0,T)$$ and $$R > 0$$, set $${\it {\Gamma}}(t) = \max\{0,R-Mt\}$$. Suppose $$\phi \in C^1({\mathbb{R}})$$ is non-negative and satisfies $$\left|{\nabla \phi}\right| \le C_\phi\phi$$. Then there exist constants $$C_1$$ and $$C_2$$ depending only on $$p,\sigma,f,C_\phi$$ such that   E[∫B(0,Γ(tn))|un(x)|pϕ(x)dx] ≤eC1tnE[∫B(0,R)|u0(x)|pϕ(x)dx] +C2tneC1tn∫B(0,R)ϕ(x)dx. (4.1) If $$\sigma(x,0) = 0$$ then $$C_2 = 0$$. Here, $$B(0,R)$$ denotes the open ball with radius $$R$$ centered at $$0$$. Remark 4.2 Suppose $$\phi \in \mathfrak{N}$$ and $$u^0 \in L^p({\it {\Omega}};L^p({\mathbb{R}}^d,\phi))$$. Then, $$\phi \in L^1({\mathbb{R}}^d)$$ and the right-hand side of (4.1) is bounded independently of $$R > 0$$. It follows that $$u^n \in L^p({\it {\Omega}};L^p({\mathbb{R}}^d,\phi))$$. Proof. 1. Deterministic step. We want to prove the following: with $$1 \le p < \infty$$, let $$v^0 \in L^p_{\mathrm{loc}}({\mathbb{R}}^d)$$ and $$v(t) = {\mathcal{S}_{\text{CL}}}(t)v^0$$. Then, for any $$0 < \tau \leq T$$,   ∫B(0,Γ(τ))|v(τ,x)|pϕ(x)dx≤e‖f‖LipCϕt∫B(0,R)|v0(x)|pϕ(x)dx. (4.2) We might as well assume $${\it {\Gamma}}(\tau) > 0$$. As $$v$$ is an entropy solution of (3.1),   ∬ΠT∫RdS(v(t,x))∂tφ+Q(v(t,x))⋅∇xφdxdt+∫RS(v0(x))φ(0,x)dx≥0 (4.3) for all non-negative $$\varphi \in C^\infty_c([0,T) \times {\mathbb{R}}^d)$$, for any convex $$S\in C^2$$ with $$S'$$ bounded and $$Q'=S'f'$$. Let $$0 < \delta < \mathrm{min}\left\{{{\it {\Gamma}}(\tau),\frac{1}{2}\tau}\right\}$$. Take   φ(t,x)=ψδ(t)Hδ(Γ(t),|x|)ϕ(x), where   ψδ(t)=1−∫0tJδ+(τ−ζ)dζ and Hδ(L,r)=∫−δLJδ(ζ−r)dζ. If $$\phi \in C^\infty({\mathbb{R}}^d)$$ then $$\varphi$$ is a non-negative function in $$C^\infty_c([0,T) \times {\mathbb{R}}^d)$$. To see this, note that by assumption, $${\it {\Gamma}}(t) > \delta$$ for all $$t \in \mathrm{supp}(\psi_\delta) \subset [0,\tau)$$. Hence, restricted to the support of $$\psi_\delta$$, $${\it {\Gamma}}(t) = R-Mt$$. Furthermore, $$H_\delta({\it {\Gamma}}(t),\left|{x}\right|) = 1$$ for all $$x \in B(0,{\it {\Gamma}}(t)-\delta)$$. By approximation, it suffices with $$\phi \in C^1({\mathbb{R}}^d)$$ for (4.3) to hold true. Recall that $$\frac{d}{\mathrm{d}t} {\it {\Gamma}}(t) = -M$$ for all $$0 \leq t \leq \tau$$ and observe that   ∂tφ(t,x) =−Jδ+(τ−t)Hδ(Γ(t),|x|)ϕ(x)−Mψδ(t)Jδ(Γ(t)−|x|)ϕ(x),∇φ(t,x) =−ψδ(t)Jδ(Γ(t)−|x|)x|x|ϕ(x)+ψδ(t)Hδ(Γ(t),|x|)∇ϕ(x). Hence,   ∫RS(v0(x))Hδ(R,|x|)ϕ(x)dx ≥∬ΠTS(v(t,x))Jδ+(τ−t)Hδ(Γ(t),|x|)ϕ(x)dxdt +∬ΠT(Q(v(t,x))⋅x|x|+MS(v(t,x)))ψδ(t)Jδ(Γ(t)−|x|)ϕ(x)dxdt⏟T1 −∬ΠTQ(v(t,x))ψδ(t)Hδ(Γ(t),|x|)⋅∇ϕ(x)dxdt⏟T2. (4.4) Suppose $$S'(0) = S(0) = 0$$. Then,   |Q(v)|=|∫0vS′(z)f′(z)dz|≤‖f‖LipS(v). It follows, as $$M \ge \left\Vert{f}\right\Vert_\mathrm{Lip}$$, that $${\mathscr{T}}^1 \ge 0$$. Because of the assumption on $$\phi$$,   |T2|≤‖f‖LipCϕ∬ΠTS(v)ψδ(t)Hδ(R,|x|)ϕ(x)dxdt. Sending $$\delta \downarrow 0$$, inequality (4.4) then takes the form   X(τ)≤X(0)+‖f‖LipCϕ∫0τX(r)dr, where   X(t)=∫B(0,Γ(t))S(v(t,x))ϕ(x)dx. Next, apply Grönwall’s inequality. Estimate (4.2) follows upon letting $$S \rightarrow \left|{\cdot}\right|^p$$ and applying the dominated convergence theorem. 2. Stochastic step. We want to prove the following. Fix $$2 \le p < \infty$$. Suppose $$w(s) \in L^p({\it {\Omega}},{\mathscr{F}}_s,P;L^p_\mathrm{loc}({\mathbb{R}}))$$ and take $$w(t) = {\mathcal{S}_{\text{SDE}}}(t,s)w(s)$$ for $$s \leq t$$. For any $$R > 0,$$ there exist constants $$C_3$$ and $$C_2$$ depending only on $$p$$ and $$\sigma$$ such that   E[∫B|w(t,x)|pϕ(x)dx]≤eC3(t−s)(E[∫B|w(s,x)|pϕ(x)dx]+C2(t−s)∫Bϕ(x)dx). (4.5) If $$\sigma(x,0) = 0$$ then $$C_2 = 0$$. By Itô’s lemma,   dS(w)=12S′′(w)σ(x,w)2dt+S′(w)σ(x,w)dB for any $$S \in C^2$$. Without loss of generality, we can assume $$p=2,4,6,\,{\ldots}\,.$$ Taking $$S(u)=\left|{u}\right|^p$$, multiplying by $$\phi$$ and integrating over $$B = B(0,R)$$, we arrive at   E[∫B|w(t,x)|pϕ(x)dx]−E[∫B|w(s,x)|pϕ(x)dx] ≤p(p−1)2∫stE[∫Bw(r,x)p−2σ(x,w(r,x))2ϕ(x)dx]dr. Recall that $$\sigma(x,w) \le \left|{\sigma(x,0)}\right| + \left\Vert{\sigma}\right\Vert_{\mathrm{Lip}}\left|{w}\right|$$. Hence, according to assumption (𝓐σ),   T3 :=p(p−1)2E[∫Bw(r,x)p−2σ(x,w(r,x))2ϕ(x)dx] ≤p(p−1)(‖σ(⋅,0)‖∞2E[∫B|w(r,x)|p−2ϕ(x)dx]  +‖σ‖Lip2E[∫B|w(r,x)|pϕ(x)dx]). Applying Hölder’s inequlity with $$\theta = \frac{p}{p-2}$$ and $$\theta' = \frac{p}{2}$$,   ∫B(|w(r,x)|pϕ(x))1/θϕ(x)1/θ′⏟|w(r,x)|p−2ϕ(x)dx≤(∫B|w(r,x)|pϕ(x)dx)1/θ⏟A(∫Bϕ(x)dx)1/θ′⏟B. Because of Young’s inequality $$AB \leq \frac{1}{\theta}A^\theta + \frac{1}{\theta'}B^{\theta'}$$. It follows that   ∫B|w(r,x)|p−2ϕ(x)dx≤p−2p∫B|w(r,x)|pϕ(x)dx+2p∫Bϕ(x)dx. Consequently,   T3 ≤(p−1)((p−2)‖σ(⋅,0)‖∞+p‖σ‖Lip2)⏟C3E[∫B|w(r,x)|pϕ(x)dx] +2(p−1)‖σ(⋅,0)‖∞2⏟C2∫Bϕ(x)dx. It follows that   E[∫B|w(t,x)|pϕ(x)dx] ≤E[∫B|w(s,x)|pϕ(x)dx] +C3∫stE[∫B|w(r,x)|pϕ(x)dx]dr+C2(∫Bϕ(x)dx)(t−s). This inequality is of the general form   X(t)≤X(s)+∫stK(r)X(r)dr+∫stH(r)dr. (4.6) Appealing to Grönwall’s inequality,   X(t)≤exp⁡[∫stK(r)dr]X(s)+∫stexp⁡[∫rtK(u)du]H(r)dr. (4.7) Identifying $$K=C_3$$ and $$H=C_2\left\Vert{\phi}\right\Vert_{L^1(B)}$$, it follows that   E[∫B|w(t,x)|pϕ(x)dx]≤eC3(t−s)E[∫B|w(s,x)|pϕ(x)dx]+C2‖ϕ‖L1(B)∫steC3(t−r)dr. Next, observe that $$e^{C_3(t-r)} \leq e^{C_3(t-s)}$$ for all $$s \leq r \leq t$$, and so (4.5) follows. 3. Inductive step. Let $$P_n$$ be the statement that (4.1) is true, and note that $$P_0$$ is trivially true. We must show that $$P_n$$ implies $$P_{n+1}$$. By (3.3), $$u^{n+1} = {\mathcal{S}_{\text{SDE}}}(t_{n+1},t_n){\mathcal{S}_{\text{CL}}}({{{\it {\Delta}} t}})u^n$$. Recall that $${v_{{{{\it {\Delta}} t}}}}((t_{n+1})-) = {\mathcal{S}_{\text{CL}}}({{{\it {\Delta}} t}})u^n$$. By (4.2),   E[∫B(0,Γ(tn+1))|vΔt((tn+1)−,x)|pϕ(x)dx]≤e‖f‖LipCϕΔtE[∫B(0,Γ(tn))|un(x)|pϕ(x)dx]. Because $$u^{n+1} = {\mathcal{S}_{\text{SDE}}}(t_{n+1},t_n){v_{{{{\it {\Delta}} t}}}}((t_{n+1})-),$$ it follows from (4.5) that   E[∫B(0,Γ(tn+1))|un+1(x)|pϕ(x)dx] ≤eC3Δt(E[∫B(0,Γ(tn+1))|vΔt((tn+1)−,x)|pϕ(x)dx]+C2∫B(0,Γ(tn+1))ϕ(x)dxΔt). Combining the two previous estimates,   E[∫B(0,Γ(tn+1))|un+1(x)|pϕ(x)dx] ≤eC3Δt(e‖f‖LipCϕΔtE[∫B(0,Γ(tn))|un(x)|pϕ(x)dx] +C2Δt∫B(0,Γ(tn+1))ϕ(x)dx) ≤eC1Δt(E[∫B(0,Γ(tn))|un(x)|pϕ(x)dx] +C2Δt∫B(0,R)ϕ(x)dx),C1=‖f‖LipCϕ+C3. Inserting the induction hypothesis brings to an end the proof of (4.1). □ Corollary 4.3 Let $${u_{{{{\it {\Delta}} t}}}}$$ and $${v_{{{{\it {\Delta}} t}}}}$$ be defined by (3.4) and (3.5), respectively, and suppose $$u^0$$ belongs to $$L^q({\it {\Omega}},{\mathscr{F}}_0,P;L^q({\mathbb{R}}^d,\phi))$$, $$2 \le q < \infty$$, $$\phi \in \mathfrak{N}$$. Then, for each $$1 \le p \le q$$, there exists a finite constant $$C$$ independent of $${{{\it {\Delta}} t}}$$ (but dependent on $$T,p,\phi,f,\sigma,u^0$$) such that   max{E[‖uΔt(t)]‖p,ϕp,E[‖vΔt(t)]‖p,ϕp}≤C,t∈[0,T]. Proof. It suffices to prove the result for $$p = q$$. To this end, suppose $$1 \le p < q$$ and $$w \in L^q({\mathbb{R}}^d,\phi)$$. Let $$r = q/p$$, $$r' = q/(q-p)$$, so that $$\frac{1}{r}+ \frac{1}{r'} = 1$$. Take $$f = \left|{u}\right|^p\phi^{1/r}$$, $$g = \phi^{1/r'}$$ and apply Hölder’s inequality. The result is   ∫Rd|w(x)|pϕ(x)dx≤(∫Rd|w(x)|qϕ(x)dx)p/q(∫Rdϕ(x)dx)1−p/q. (4.8) Consider the case $$p = q$$. By Proposition 4.1, there exists a constant $$C > 0$$ depending only on $$q,f,\sigma,u^0,T,\phi$$ such that   E[‖un‖q,ϕq]≤C,0≤n≤N. Let $$t \in [t_n, t_{n+1})$$. By (4.2),   E[‖SCL(t−tn)un⏟vΔt(t)‖q,ϕq]≤e‖f‖LipCϕΔtE[‖un‖q,ϕq]. This finishes the proof for $${v_{{{{\it {\Delta}} t}}}}$$. For $${u_{{{{\it {\Delta}} t}}}}$$, the result follows by (4.5). □ The next result should be compared with Karlsen & Storrøsten (2017, Proposition 5.2) and Chen et al. (2012, Section 6). It can be turned into a fractional $$BV_x$$ estimate ($$L^1$$-space translation estimate) along the lines of Chen et al. (2012), but we will not need this fact here. Proposition 4.4 (Fractional $$BV_x$$ estimates). Suppose (𝓐f), (𝓐f,1), (𝓐σ) and (𝓐σ,1) are satisfied. Let $$\phi \in \mathfrak{N}$$. Suppose $$u^0 \in L^2({\it {\Omega}},{\mathscr{F}}_0,P;L^2({\mathbb{R}}^d,\phi))$$. Let $$u_{{{\it {\Delta}} t}}$$ and $${v_{{{{\it {\Delta}} t}}}}$$ be defined by (3.4) and (3.5), respectively. Then there exists a constant $$C_{T}$$, independent of $${{{\it {\Delta}} t}}$$, such that   [∬Rd×Rd|uΔt(t,x+z)−uΔt(t,x−z)|Jr(z)ϕ(x)dxdz] ≤eCϕ‖f‖Lipt[∬Rd×Rd|u0(x+z)−u0(x−z))|Jr(z)ϕ(x)dxdz]+CTrκσ for any $$t\in (0,T)$$. Here $${{\kappa_\sigma}}\in (0,1/2]$$ is defined in (𝓐σ,1). If $$\sigma(x,u) = \sigma(u)$$ then we may take $$C_{T} = 0$$. The same result holds with $${u_{{{{\it {\Delta}} t}}}}$$ replaced by $${v_{{{{\it {\Delta}} t}}}}$$. Remark 4.5 In the deterministic case or whenever $$\sigma=\sigma(u)$$ is independent of the spatial location $$x$$, we recover the usual BV bound. To this end, note that $$C_T = 0$$, apply the weight $$\phi_\rho(x)=\exp{-\rho\sqrt{1+\left|{x}\right|^2}}$$ ($$\rho > 0$$) and then send $$\rho \downarrow 0$$. Before we proceed to the proof, we fix some notation and make a few observations. Let us define $$C^2$$-approximations $$\left\{{S_\delta}\right\}_{\delta > 0}$$ of the absolute value function by asking that   Sδ′(σ)=2∫0σJδ(z)dz,Sδ(0)=0. (4.9) Then,   |r|−δ≤Sδ(r)≤|r|,|Sδ″(r)|≤2δ‖J‖∞1|r|<δ. (4.10) Given $$S_\delta$$, we define $$Q_\delta$$ by   Qδ(u,v)=∫vuSδ′(ξ−v)f′(ξ)dξ,u,v∈R. (4.11) This function satisfies   |∂u(Qδ(u,v)−Qδ(v,u))|≤‖f′′‖L∞δ (4.12) and   |Qδ(u,v)|≤‖f‖LipSδ(u−v). (4.13) Let us state two convenient identities. First, for $$h = h(\cdot,\cdot)\in L^1_{{{\mathrm{loc}}}}$$,   12d∬Rd×Rdh(x,y)ϕ(x+y2)Jr(x−y2)dxdy=∬Rd×Rdh(x~+z,x~−z)ϕ(x~)Jr(z)dx~dz. (4.14) This follows by a change of variables: $$(\tilde{x},z) = \left(\frac{x+y}{2}, \frac{x-y}{2}\right)$$, $$\mathrm{d}y = 2^d \mathrm{d}z$$. Next,   12d∫Rdϕ(x+y2)Jr(x−y2)dy=(ϕ⋆Jr)(x). (4.15) Proof of Proposition 4.4. Given $$u=u(t)=u(t,x;\omega)$$, we introduce the quantity   Dru(t):=[12d∬Rd×Rd|u(t,x)−u(t,y)|Jr(x−y2)ϕ(x+y2)dxdy]. Actually, at first we are not going to work with this quantity but rather with   Dr,δu(t):=[12d∬Rd×RdSδ(u(t,x)−u(t,y))Jr(x−y2)ϕ(x+y2)dxdy], where the regularized entropy $$S_\delta$$ is defined in (4.9). In view of (4.10) and (4.15),   |Dru(t)−Dr,δu(t)|≤‖ϕ‖L1(Rd)δ,t>0. (4.16) 1. Deterministic step. Let $$v(t,x)$$ be the unique entropy solution of (3.1). We want to prove the following claim: there exists a constant $$C_1$$ depending only on $$J$$ and $$C_\phi$$ such that for all $$0 < r \le 1$$,   Dr,δv(t)≤eCϕ‖f‖Lipt(Dr,δv(0)+C1‖f″‖∞E[‖v0‖1,ϕ]t(δr)). (4.17) Let $$Q_\delta$$ be defined in (4.11). Using the entropy inequalities and Kružkov’s method of doubling the variables, it follows in a standard way that for $$t>0$$,   12d∬Rd×RdSδ(v(t,x)−v(t,y))Jr(x−y2)ϕ(x+y2)dxdy −12d∬Rd×RdSδ(v0(x)−v0(y))Jr(x−y2)ϕ(x+y2)dxdy ≤12d∫0t∬Rd×RdQδ(v(s,x),v(s,y))⋅∇ϕ(x+y2)Jr(x−y2)dxdyds +12d∫0t∬Rd×Rd(Qδ(v(s,y),v(s,x))−Qδ(v(s,x),v(s,y)))⋅∇y(ϕ(x+y2)Jr(x−y2))dxdyds =:TCL1+TCL2. By (4.13),   |TCL1|≤Cϕ‖f‖Lip12d∫0t∬Rd×RdSδ(v(s,x)−v(s,y))Jr(x−y2)ϕ(x+y2)dxdyds. Consider $${\mathscr{T}}_{\text{CL}}^2$$. Thanks to (4.12),   |Qδ(v,u)−Qδ(u,v)|=|∫vu∂ξ(Qδ(ξ,v)−Qδ(v,ξ))dξ|≤‖f″‖∞|u−v|δ, so that   |TCL2| ≤‖‖f″‖∞‖2δ12d∫0t∬Rd×Rd|v(s,x)−v(s,y)||∇Jr(x−y2)|ϕ(x+y2)dxdyds +‖f″‖∞2δ12d∫0t∬Rd×Rd|v(s,x)−v(s,y)|Jr(x−y2)|∇ϕ(x+y2)|dxdyds =:TCL2,1+TCL2,2. Consider $${\mathscr{T}}_{\text{CL}}^{2,1}$$. Setting $$\varphi_r(z) = \left\Vert{\nabla J}\right\Vert_1^{-1}\frac{1}{r^d} \left|{\nabla J(\frac{z}{r})}\right|$$, we write   |∇Jr(x−y2)|=‖∇J‖11rφr(x−y2). By the triangle inequality and (4.15),   12d∬Rd×Rd|v(s,x)−v(s,y)||∇Jr(x−y2)|ϕ(x+y2)dxdy ≤‖∇J‖12r∫Rd|v(s,x)|(ϕ⋆φr)(x)dx=‖∇J‖12r‖v(s)‖1,ϕ⋆φr. Considering $${\mathscr{T}}_{\text{CL}}^{2,2}$$, with $$\phi \in \mathfrak{N}$$,   12d∬Rd×Rd|v(s,x)−v(s,y)|Jr(x−y2)|∇ϕ(x+y2)|dxdy ≤2Cϕ∫Rd|v(s,x)|(ϕ⋆Jr)(x)dx=2Cϕ‖v(s)‖1,ϕ⋆Jr. By Lemma A.8,   max{‖v(s)‖1,ϕ⋆φr,‖v(s)‖1,ϕ⋆Jr}≤‖v(s)‖1,ϕ(1+w1,ϕ(r)), where $$w_{1,\phi}$$ is defined in Lemma A.7. Hence,   |TCL2|≤‖f″‖∞(1+w1,ϕ(r))(∫0t‖v(s)‖1,ϕds)(‖∇J‖11r+Cϕ)δ. In view of (4.2), $$\left\Vert{v(s)}\right\Vert_{1,\phi} \le e^{\left\Vert{f}\right\Vert_{\mathrm{Lip}}C_\phi s}\left\Vert{v_0}\right\Vert_{1,\phi}$$. Summarizing,   12d∬Rd×RdSδ(v(t,x)−v(t,y))Jr(x−y2)ϕ(x+y2)dxdy −12d∬Rd×RdSδ(v0(x)−v0(y))Jr(x−y2)ϕ(x+y2)dxdy ≤Cϕ‖f‖Lip⏟K∫0t12d∬Rd×RdSδ(v(s,x)−v(s,y))Jr(x−y2)ϕ(x+y2)dxdyds +∫0tC1‖f″‖∞‖v0‖1,ϕe‖f‖LipCϕs(δr)⏟H(s)ds, (4.18) where $$C_1 = (1+w_{1,\phi}(1))(\left\Vert{\nabla J}\right\Vert_1 + C_\phi) $$. This inequality is of the form (4.6). By Grönwall’s inequality (4.7),    ∬Rd×RdSδ(v(t,x)−v(t,y))Jr(x−y2)ϕ(x+y2)dxdy ≤eCϕ‖f‖Lipt(  ∬Rd×RdSδ(v0(x)−v0(y))Jr(x−y2)ϕ(x+y2)dxdy  +C1‖f″‖∞‖v0‖1,ϕt(δr)∬Rd×Rd). This proves the claim (4.17). 2. Stochastic step. Let $$w(t)={\mathcal{S}_{\text{SDE}}}(t,s)w(s)$$. We will now derive an estimate for $$w$$ similar to (4.18). There exist constants $$C_1$$ and $$C_2$$, depending only on $$J,\sigma,\phi$$, such that   Dr,δw(t)≤Dr,δw(s)+C1r2κ+1δ∫stE[‖1+|w(τ)|‖2,ϕ2]dτ+C2(t−s)δ, (4.19) for all $$0 \le r \le 1$$. If $$M_{\sigma} = 0$$ then $$C_1 = 0$$. Since $$w(t,x)-w(t,y)$$ solves   d(w(t,x)−w(t,y))=(σ(x,w(t,x))−σ(y,w(t,y)))dB(t) applying Itô’s formula to $$S_\delta(w(t,x)-w(t,y))$$ yields   dSδ(w(t,x)−w(t,y)) =12Sδ′′(w(t,x)−w(t,y))(σ(x,w(t,x))−σ(y,w(t,y)))2dt, +Sδ′(w(t,x)−w(t,y))(σ(x,w(t,x))−σ(y,w(t,y)))dB(t). Integrating against the test function $$\frac{1}{2^d}J_r(\tfrac{x-y}{2})\phi(\tfrac{x+y}{2})$$, we arrive at   12d∬Rd×RdSδ(w(t,x)−w(t,y))Jr(x−y2)ϕ(x+y2)dxdy −12d∬Rd×RdSδ(w(s,x)−w(s,y))Jr(x−y2)ϕ(x+y2)dxdy =∫st12d∬Rd×Rd12Sδ′′(w(τ,x)−w(τ,y)) ×(σ(x,w(τ,x))−σ(y,w(τ,y)))2Jr(x−y2)ϕ(x+y2)dxdydτ +∫st12d∬Rd×RdSδ′(w(τ,x)−w(τ,y))(σ(x,w(τ,x))−σ(y,w(τ,y)))dxdydB(τ) =TSDE1+TSDE2, where the $${\mathscr{T}}_{\text{SDE}}^2$$ term has zero expectation. Note that   (σ(x,u)−σ(y,v))2≤2(σ(x,u)−σ(x,v))2+2(σ(x,v)−σ(y,v))2 for any $$u,v \in {\mathbb{R}}$$. We estimate the $${\mathscr{T}}_{\text{SDE}}^1$$ term as follows:   E[|TSDE1|] ≤2E[∫st12d∬Rd×RdJδ(w(τ,x)−w(τ,y))(σ(x,w(τ,x))−σ(y,w(τ,x)))2  ×Jr(x−y2)ϕ(x+y2)dxdydτ\raisebox25pt ] +2E[∫st12d∬Rd×RdJδ(w(τ,x)−w(τ,y))(σ(y,w(τ,x))−σ(y,w(τ,y)))2  ×Jr(x−y2)ϕ(x+y2)dxdydτ\raisebox25pt ]=:S1+S2. Regarding $$S_1$$, recall that $$\left|{J_\delta}\right| \le \left\Vert{J}\right\Vert_\infty/\delta$$. By (𝓐σ,1),   |S1| ≤‖J‖∞2δE[∫st12d∬Rd×Rd|σ(x,w(τ,x))−σ(y,w(τ,x))|2×Jr(x−y2)ϕ(x+y2)dxdydτ] ≤‖J‖∞Mσ22δE[∫st12d∬Rd×Rd|x−y|2κσ+1(1+|w(τ,x)|)2×Jr(x−y2)ϕ(x+y2)dxdydτ] ≤2‖J‖∞Mσ2(2r)2κσ+1δ∫stE[‖1+|w(τ)|‖2,ϕ⋆Jr2]dτ. By Lemma A.8,   ‖1+|w(τ)|‖2,ϕ⋆Jr2≤‖1+|w(τ)|‖2,ϕ2(1+w1,ϕ(r)), where $$w_{1,\phi}$$ is defined in Lemma A.7. It follows that   |S1|≤22(κσ+1)‖J‖∞Mσ2(1+w1,ϕ(1))⏟C1∫stE[‖1+|w(τ)|‖2,ϕ2]dτr2κσ+1δ for all $$0 < r \leq 1$$. Consider $$S_2$$. Because of assumption (𝓐σ),   Jδ(w(τ,x)−w(τ,y))(σ(y,w(τ,x))−σ(y,w(τ,y)))2≤‖σ‖Lip2‖J‖∞δ. Hence,   |S2|≤2‖σ‖Lip2‖J‖∞‖ϕ‖L1(Rd)⏟C2(t−s)δ. This proves (4.19) 3. Inductive step. Let $$P_n$$ be the following claim: there exist constants $$C_1,C_2,C_3$$ depending only on $$J,\phi,\sigma$$ such that for all $$0 < r \le 1$$,   Dr,δun ≤eCϕ‖f‖Liptn(Dr,δu0+C3‖f″‖∞(Δt∑k=0n−1E[‖uk‖1,ϕ])δr  +C1r2κσ+1δ∫0tnE[‖1+|uΔt(t)|‖2,ϕ2]dt+C2tnδ). (4.20) If $$M_{\sigma} = 0$$ then $$C_1 = 0$$. Note that $$P_0$$ is trivially true. Assuming that $$P_n$$ is true, we want to verify $$P_{n+1}$$. Recall that $$u^{n+1} = {\mathcal{S}_{\text{SDE}}}(t_{n+1},t_n){\mathcal{S}_{\text{CL}}}({{{\it {\Delta}} t}})u^n$$. Let $$w^n = {\mathcal{S}_{\text{CL}}}({{{\it {\Delta}} t}})u^n$$ and note that $${\mathcal{S}_{\text{SDE}}}(t,t_n)w^n = u_{{{\it {\Delta}} t}}(t)$$ for $$t_n \le t <t_{n+1}$$. As $$1 \le e^{C_\phi\left\Vert{f}\right\Vert_{\mathrm{Lip}}{{{\it {\Delta}} t}}}$$, (4.19) yields   Dr,δun+1≤Dr,δwn+eCϕ‖f‖LipΔt(C1r2κσ+1δ∫tntn+1E[‖1+|uΔt(t)|‖2,ϕ2]dt+C2Δtδ). By (4.17),   Dr,δwn≤eCϕ‖f‖LipΔt(Dr,δun+C3‖f″‖∞E[‖un‖1,ϕ]Δt(δr)). Hence,   Dr,δun+1 ≤eCϕ‖f‖LipΔt(Dr,δun+C3‖f″‖∞E[‖un‖1,ϕ]Δt(δr) +C1r2κσ+1δ∫tntn+1E[‖1+|uΔt(t)|‖2,ϕ2]dt+C2Δtδ) and inserting the hypothesis $$P_n$$ yields $$P_{n+1}$$. 4. Concluding the proof. Consider (4.20). By Corollary 4.3, there exists a constant $$C$$, independent of $${{{\it {\Delta}} t}}$$, such that   Dr,δun≤eCϕ‖f‖Liptn(Dr,δu0+Ctn(δr+δ+r2κσ+1δ)). Because of (4.16), this translates into   Drun≤eCϕ‖f‖Liptn(Dru0+Ctn(δr+δ+r2κσ+1δ)+2‖ϕ‖L1(Rd)δ),0≤n≤N. We can argue via (4.19) to obtain   DruΔt(t)≤eCϕ‖f‖Lipt(Dru0+Ct(δr+δ+r2κσ+1δ)+2‖ϕ‖L1(Rd)δ),t∈[0,T]. Note that the same holds true if we replace $${u_{{{{\it {\Delta}} t}}}}$$ by $${v_{{{{\it {\Delta}} t}}}}$$, thanks to (4.17). Viewing $$r > 0$$ as fixed, we can choose $$\delta = r^{{{\kappa_\sigma}} +1}$$ to arrive at the bound   DruΔt(t)≤eCϕ‖f‖LiptDru0+CTrκσ. The result follows by (4.14). In the case that $$M_{\sigma} = 0$$, we have   DruΔt(t)≤eeCϕ‖f‖LiptDru0+CT(δr+δ),t∈(0,T), and we may send $$\delta \downarrow 0$$ independently of $$r$$. □ In Proposition 4.4, the spatial regularity of $${u_{{{{\it {\Delta}} t}}}}, {v_{{{{\it {\Delta}} t}}}}$$ is characterized in terms of averaged $$L^1$$-space translates. In the BV context, this is equivalently characterized by integration against the divergence of a smooth bounded function. Restricting to one dimension ($$d = 1$$) and $$u \in C^1({\mathbb{R}})$$, we have   suph>0{1h∫R|u(x+h)−u(x)|dx}=∫R|u′(x)|dx=sup{∫Ru(x)β′(x)dx:β∈Cc∞(R),‖β‖∞≤1}. Fix $$\kappa \in (0,1]$$. The left-hand side has a natural generalization to the fractional BV setting by considering $$u \in L^1({\mathbb{R}})$$ satisfying   suph>0{1hκ∫R|u(x+h)−u(x)|dx}<∞. (4.21) A possible generalization of the right-hand side reads   sup{δ1−κ∫Ru(x)(Jδ⋆β)′(x)dx:δ>0,‖β‖∞≤1}<∞, (4.22) where $$\left\{{J_\delta}\right\}_{\delta > 0}$$ is a suitable family of symmetric mollifiers. The next lemma shows that (4.22) may be bounded in terms of (4.21). The lemma plays a key role in obtaining the optimal $$L^1$$-time continuity estimates in Proposition 4.8. Lemma 4.6 Let $$\rho \in C^\infty_c((0,1))$$ satisfy $$\int_0^1 \rho(r)\,\mathrm{d}r = 1$$ and $$\rho \geq 0$$. For $$x \in {\mathbb{R}}^d$$ define   U(x)=1α(d)Md(1−∫0|x|ρ(r)dr),V(x)=1dα(d)Md−1ρ(|x|), where $$M_n = \int_0^\infty r^n\rho(r)\,\mathrm{d}r$$, $$n \geq 0$$ and $$\alpha(d)$$ denotes the volume of the unit ball in $${\mathbb{R}}^d$$. Then $$U,V$$ are symmetric mollifiers on $${\mathbb{R}}^d$$ with support in $$B(0,1)$$. For $$\phi \in \mathfrak{N}$$, $$u \in L^1({\mathbb{R}}^d,\phi)$$ and $$\delta > 0$$, define   Vδ(u)=∬Rd×Rd|u(x+z)−u(x−z)|Vδ(z)ϕ(x)dzdx, where $$V_\delta(z) = \delta^{-d}V(\delta^{-1}z)$$. Similarly, for $$\beta \in L^\infty({\mathbb{R}}^d)$$ let   Uδi(u,β)=∫Rdu(x)∂xi(Uδ⋆β)(x)ϕ(x)dx,1≤i≤d, where $$U_\delta(z) = \delta^{-d}U(\delta^{-1}z)$$. Then   |Uδi(u,β)|≤dMd−12Md(1δVδ(u)+2‖u‖1,ϕw1,ϕ(δ)δ)‖β‖L∞, for each $$1 \leq i \leq d$$, where $$w_{1,\phi}$$ is defined in Lemma A.7. Remark 4.7 We note that Lemma 4.6 covers the BV case. If there is a constant $$C \geq 0$$ such that $$\mathcal{V}_\delta(u) \leq C \delta$$ (the BV case) then   ∫Rdu(∇⋅β)ϕdx=limδ↓0∑i=1dUδi(u,βi)≤d2Md−12Md(C+2Cϕ‖u‖1,ϕ) for any $$\beta = (\beta^1, \dots,\beta^d) \in C^1_c({\mathbb{R}}^d;{\mathbb{R}}^d)$$ satisfying $$\left\Vert{\beta}\right\Vert_\infty \leq 1$$. It follows that   ∫Rd|∇u|ϕdx ≤sup|β|≤1∫Rd(∇u⋅β)ϕdx =sup|β|≤1∫Rdu(∇⋅β)ϕ+u(β⋅∇ϕ)dx ≤d2Md−12Md(C+2Cϕ‖u‖1,ϕ)+Cϕ‖u‖1,ϕ and so $$\left|{\nabla u}\right|$$ is a finite measure with respect to $$\phi\, \mathrm{d}x$$. Proof. Let us first show that $$U$$ is a symmetric mollifier. It is clearly symmetric, furthermore it is smooth since $$\left\{{0}\right\} \notin \mathrm{cl}(\mathrm{supp}(\rho))$$. Change to polar coordinates and integrate by parts to obtain   ∫Rd(1−∫0|x|ρ(σ)dσ)dx =α(d)∫0∞drd−1(1−∫0rρ(σ)dσ)dr =α(d)∫0∞rdρ(r)dr=α(d)Md. Similarly for $$V$$,   ∫Rdρ(|x|)dx=dα(d)∫0∞rd−1ρ(r)dr=dα(d)Md−1. Note that   Uδi(u,β)=∬Rd×Rdu(x)∂xiUδ(x−y)β(y)ϕ(x)dydx. Next, we differentiate to obtain   ∂xiUδ(x)=−1α(d)Md1δdρ(|x|δ)1δsign(xi)=−dMd−1MdVδ(x)1δsign(xi). Hence,   Uδi(u,β)=−dMd−1Md1δ∬Rd×Rdu(x)Vδ(x−y)sign(xi−yi)β(y)dydx. This integral may be reformulated according to    ∬Rd×Rdu(x)Vδ(x−y)sign(xi−yi)β(y)ϕ(x)dydx =12∬Rd×Rdu(x)Vδ(x−y)sign(xi−yi)β(y)ϕ(x)dydx −12∬Rd×Rdu(x)Vδ(y−x)sign(yi−xi)β(y)ϕ(x)dydx =12∬Rd×Rd(u(y−z)ϕ(y−z)−u(y+z)ϕ(y+z))Vδ(z)sign(zi)β(y)dzdy, where we made the substitution $$x = y-z$$ and $$x = y + z,$$ respectively. Since   u(y−z)ϕ(y−z)−u(y+z)ϕ(y+z) =(u(y−z)−u(y+z))ϕ(y) +u(y−z)(ϕ(y−z)−ϕ(y))−u(y+z)(ϕ(y+z)−ϕ(y)), it follows that   Uδi(u,β) =dMd−12Md1δ∬Rd×Rd(u(y+z)−u(y−z))Vδ(z)sign(zi)β(y)ϕ(y)dzdy +dMd−12Md1δ∬Rd×Rdu(y+z)(ϕ(y+z)−ϕ(y))Vδ(z)sign(zi)β(y)dzdy +dMd−12Md1δ∬Rd×Rdu(y−z)(ϕ(y)−ϕ(y−z))Vδ(z)sign(zi)β(y)dzdy =:Zδ1+Zδ2+Zδ3. Clearly,   |Zδ1|≤dMd−12Md1δ∬Rd×Rd|u(y+z)−u(y−z)|Vδ(z)ϕ(y)dzdy⏟Vδ(u)‖β‖L∞. Consider $${\mathscr{Z}}_\delta^2$$; the term $${\mathscr{Z}}_\delta^3$$ is treated similarly. By Lemma A.7,   |ϕ(y+z)−ϕ(y)|≤w1,ϕ(|z|)ϕ(y+z). Hence, by Young’s inequality for convolutions,   |Zδ2|≤dMd−12Mdw1,ϕ(δ)δ∫Rd(|uϕ|⋆Vδ)(y)dy‖β‖L∞≤dMd−12Mdw1,ϕ(δ)δ‖u‖1,ϕ‖β‖L∞. This concludes the proof of the lemma. □ Next, we consider the time continuity of the splitting approximations. Recall that the interpolants $${u_{{{{\it {\Delta}} t}}}},{v_{{{{\it {\Delta}} t}}}}$$ are discontinuous at $$t_n = n{{{\it {\Delta}} t}}$$. Hence, the result must somehow quantify the size of the jumps as $${{{\it {\Delta}} t}} \downarrow 0$$. The idea of the proof is to ‘transfer à la Kružkov’ spatial regularity to temporal continuity Kružkov (1969, 1970). Given a bounded variation bound, or some spatial $$L^1$$-modulus of continuity, this approach has been applied to miscellaneous splitting methods for deterministic problems; cf. Holden et al. (2010) (and references therein). At variance with Holden et al. (2010), we quantify spatial regularity differently, namely in terms of averaged (weighted) $$L^1$$-translates. Combined with Lemma 4.6, we deduce $$L^1$$-time continuity estimates that recover the optimal estimates in the $$BV_x$$ case ($$\kappa=1$$). Proposition 4.8 ($$L^1$$-time continuity). Assume that (𝓐f), (𝓐f,1), (𝓐σ) and (𝓐σ,1) hold. Fix $$\phi \in \mathfrak{N}$$, and let $$u^0 \in L^2({\it {\Omega}},{\mathscr{F}}_0,P;L^2({\mathbb{R}}^d,\phi))$$ satisfy   E[∬Rd×Rd|u0(x+z)−u0(x−z))|Jr(z)ϕ(x)dxdz]=O(rκ0) (4.23) for any symmetric mollifier $$J$$ and some $$0 < {{\kappa_0}} \leq 1$$. Set   κ:={min{κ0,κσ} if σ=σ(x,u),κ0 if σ=σ(u).  Let $${u_{{{{\it {\Delta}} t}}}}$$ and $${v_{{{{\it {\Delta}} t}}}}$$ be defined in (3.4) and (3.5), respectively. (i) Suppose $$0 < \tau_1 < \tau_2 \le T$$ satisfy $$\tau_1\in (t_k,t_{k+1}]$$ and $$\tau_2\in (t_l,t_{l+1}]$$. Then there exists a finite constant $$C_{T,\phi}$$, independent of $${{{\it {\Delta}} t}}$$, such that   E[∫Rd|uΔt(τ2,x)−uΔt(τ1,x)|ϕ(x)dx]≤CT,ϕ(|(l−k)Δt|κ+τ2−τ1). (ii) Suppose $$0 \le \tau_1 \le \tau_2 < T$$ satisfy $$\tau_1\in [t_k,t_{k+1})$$ and $$\tau_2\in [t_l,t_{l+1})$$. Then there exists a finite constant $$C_{T,\phi}$$, independent of $${{{\it {\Delta}} t}}$$, such that   E[∫Rd|vΔt(τ2,x)−vΔt(τ1,x)|ϕ(x)dx]≤CT,ϕ((l−k)Δt+|τ2−τ1|κ). Proof. We shall first quantify weak continuity in the mean of $$t\mapsto {u_{{{{\it {\Delta}} t}}}}(t)$$, $$t\mapsto {v_{{{{\it {\Delta}} t}}}}(t)$$ and then turn this into fractional $$L^1$$-time continuity in the mean. The reason for first exhibiting a weak estimate is that the splitting steps do not produce functions that are Lipschitz continuous in time, thereby preventing a direct ‘inductive argument’ (see Kružkov, 1969). 1. Weak estimate. Let $$t_n = n {{{\it {\Delta}} t}}$$. Suppose $$0 < \tau_1 \le \tau_2 \le T$$ satisfies $$\tau_1\in (t_k,t_{k+1}]$$ and $$\tau_2\in (t_l,t_{l+1}]$$. Suppose $$\beta$$ belongs to $$L^\infty({\it {\Omega}} \times {\mathbb{R}}^d,{\mathscr{F}} \otimes {\mathscr{B}\left({{\mathbb{R}}^d}\right)},{\rm d}P \otimes \mathrm{d}x)$$ and let $$\beta_\delta = \beta \star U_\delta$$, where $$U_\delta$$ is defined in Lemma 4.6. We claim that there is a constant $$C > 0$$, independent of $${{{\it {\Delta}} t}}$$, such that   E[∫Rd(uΔt(τ2,x)−uΔt(τ1,x))(βδϕ)(x)dx]≤C(δκ−1(l−k)Δt+τ2−τ1)‖β‖L∞. (4.24) Consider the case $$l \geq k+1$$. We continue as follows:   T =E[∫Rd(uΔt(τ2,x)−uΔt(τ1,x))(βδϕ)(x)dx] =E[∫Rd(uΔt(τ2,x)−uΔt((tl)+,x))(βδϕ)(x)dx] +E[∫Rd(uΔt(tk+1,x)−uΔt(τ1,x))(βδϕ)(x)dx] +E[∫Rd(uΔt((tl)+,x)−uΔt(tk+1,x))(βδϕ)(x)dx] =:T1+T2+T3. Recall that $${u_{{{{\it {\Delta}} t}}}}((t_n)+) = {v_{{{{\it {\Delta}} t}}}}((t_{n+1})-) = {\mathcal{S}_{\text{CL}}}({{{\it {\Delta}} t}})u^n$$. Regarding the last term,   uΔt((tl)+,x)−uΔt(tk+1,x)=vΔt((tl+1)−,x)−vΔt(tl,x)+∑n=k+1l−1uΔt(tn+1,x)−uΔt(tn,x), where the sum is empty for the case $$l = k+1$$. Furthermore, we note that   uΔt(tn+1,x)−uΔt(tn,x)=(uΔt(tn+1,x)−uΔt((tn)+,x))+(vΔt((tn+1)−,x)−vΔt(tn,x)). This yields   T3 =E[∑n=k+1l−1∫Rd(uΔt(tn+1,x)−uΔt((tn)+,x))(βδϕ)(x)dx] +E[∑n=k+1l∫Rd(vΔt((tn+1)−,x)−vΔt(tn,x))(βδϕ)(x)dx] =E[∫Rd(∫tk+1tlσ(x,uΔt(t,x))dB(t))(βδϕ)(x)dx] +E[∑n=k+1l∫Rd(SCL(Δt)un(x)−un(x))(βδϕ)(x)dx]. It follows that $${\mathscr{T}} = {\mathscr{T}}^{\text{CL}} + {\mathscr{T}}^{\text{SDE}}$$, where   TCL :=E[∑n=k+1l∫Rd(SCL(Δt)un(x)−un(x))(βδϕ)(x)dx],TSDE :=E[∫Rd∫τ1τ2σ(x,uΔt(t,x))dB(t)(βδϕ)(x)dx]. Note that this holds true for $$k = l$$ as $${\mathscr{T}}^{\text{CL}} = 0$$ in this case. As $${v_{{{{\it {\Delta}} t}}}}(t,x)$$ is a weak solution of the conservation law (3.1) on $$[t_n,t_{n+1})$$,   |∫Rd(vΔt((tn+1)−,x)−vΔt(tn,x))(βδϕ)(x)dx| ≤|∫tntn+1∫Rdf(vΔt(r,x))⋅∇(βδϕ)(x)dxdr| ≤|∫tntn+1∫Rdf(vΔt(r,x))⋅∇βδ(x)ϕ(x)dxdr| +|∫tntn+1∫Rdf(vΔt(r,x))⋅(βδ(x)∇ϕ(x))dxdr| =:Zδ1+Zδ2. By Proposition 4.4, there exists a constant $$C > 0$$ such that   E[∬Rd×Rd|vΔt(r,x+z)−vΔt(r,x−z)]|Vδ(z)ϕ(x)dzdx≤Cδκ. Consequently, taking expectations in Lemma 4.6 yields   E[Zδ1≤d2Md−12Md‖f‖Lip(ΔtCδκ−1+2E[∫tntn+1‖vΔt(r)‖1,ϕdr]]δ−1w1,ϕ(2δ))‖β‖L∞. As $$\phi \in \mathfrak{N}$$,   Zδ2≤‖f‖LipCϕE[∫tk+1tl+1‖vΔt(t)]‖1,ϕdt‖β‖L∞. Summarizing, there exists a constant $$C$$ such that   |TCL|≤Cδκ−1(l−k)Δt‖β‖L∞ for all $$0 < \delta \le 1$$. By (4.8), Jensen’s inequality and the Itô isometry,   |TSDE| ≤‖β‖L∞∫RdE[|∫τ1τ2σ(x,uΔt(t,x))dB(t)|]ϕ(x)dx ≤‖β‖L∞‖ϕ‖L1(Rd)1/2(∫RdE[|∫τ1τ2σ(x,uΔt(t,x))dB(t)|]2ϕ(x)dx)1/2 ≤‖β‖L∞‖ϕ‖L1(Rd)1/2(∫RdE[∫τ1τ2σ2(x,uΔt(t,x))dt]ϕ(x)dx)1/2 =‖β‖L∞‖ϕ‖L1(Rd)1/2(∫τ1τ2E[‖σ(⋅,uΔt(t,⋅))]‖2,ϕ2dt)1/2 ≤C‖β‖L∞‖ϕ‖L1(Rd)1/2τ2−τ1, since, in view of (𝓐σ) and Corollary 4.3, $${E \left[{\left\Vert{\sigma(\cdot,{u_{{{{\it {\Delta}} t}}}}(t,\cdot))}\right]}\right\Vert^2_{2,\phi}}^{1/2} \leq C$$ for some constant $$C$$ independent of $$t \in [0,T]$$. Summarizing, the above estimates imply the existence of a constant $$C$$, independent of $${{{\it {\Delta}} t}},\delta$$ and $$\beta$$, such that   |T|≤C(δκ−1(k−l)Δt+τ2−τ1)‖β‖L∞, which yields (4.24). Let us consider $${v_{{{{\it {\Delta}} t}}}}$$. Suppose $$0 \le \tau_1 \le \tau_2 < T$$, with $$\tau_1 \in [t_k,t_{k+1})$$, $$\tau_2 \in [t_l,t_{l+1})$$. We claim there is a constant $$C > 0$$, independent of $${{{\it {\Delta}} t}},\delta$$ and $$\beta$$, such that   E[∫Rd(vΔt(τ2,x)−vΔt(τ1,x))(βδϕ)(x)dx]≤C(δκ−1|τ2−τ1|+(l−k)Δt)‖β‖L∞. (4.25) To prove this claim, note that   vΔt(τ2,x)−vΔt(τ1,x) =vΔt(τ2,x)−vΔt(tl,x) +∑n=k+1lvΔt(tn,x)−vΔt((tn)−,x) +∑n=k+1l−1vΔt((tn+1)−,x)−vΔt(tn,x) +vΔt((tk+1)−,x)−vΔt(τ1,x), and so   E[∫Rd(vΔt(τ2,x)−vΔt(τ1,x))(βδϕ)(x)dx]=TCL+TSDE, where   TCL :=E[∫Rd(SCL(τ2−tl)ul(x)−ul(x))(βδϕ)(x)dx] +∑n=k+1l−1E[∫Rd(SCL(Δt)un(x)−un(x))(βδϕ)(x)dx] +E[∫Rd(SCL(Δt)uk−SCL(τ1−tk)uk)(βδϕ)(x)dx],TSDE :=∑n=k+1lE[∫Rd(∫tn−1tnσ(x,uΔt(t,x))dB(t))(βδϕ)(x)dx] =E[∫Rd(∫tktlσ(x,uΔt(t,x))dB(t))(βδϕ)(x)dx]. Combining the above estimates yields (4.25). 2. Strong estimate. Let $$d(x)= {u_{{{{\it {\Delta}} t}}}}(\tau_2)-{u_{{{{\it {\Delta}} t}}}}(\tau_1)$$, $$\beta(x) = {\mathrm{sign}\left({d(x)}\right)}$$. By the triangle inequality,   E[∫Rd|uΔt(τ2,x)−uΔt(τ1,x)|ϕ(x)dx] ≤|E[∫Rdβδ(x)d(x)ϕ(x)dx]|+E[∫Rd||d(x)|−βδ(x)d(x)|ϕ(x)dx] =:T1+T2. By (4.24),   T1=O(δκ−1(l−k)Δt+τ2−τ1). Consider $${\mathscr{T}}_2$$. Following, e.g., Kružkov (1970, Lemma 1),   ||d(x)|−βδ(x)d(x)|≤∫Rd||d(x)|−d(x)sign(d(y))|Vδ(x−y)dy ≤2∫Rd|d(x)−d(y)|Vδ(x−y)dy. Upon adding and subtracting identical terms and changing variables $$2\tilde x = x+y$$, $$2z = x-y$$, it follows (after relabeling $$\tilde x$$ by $$x$$)   ∫Rd||d(x)|−βδ(x)d(x)|ϕ(x)dx ≤2∬Rd×Rd|d(x+z)−d(x−z)| ×Vδ/2(z)|ϕ(x+z)−ϕ(x)|dzdx +2∬Rd×Rd|d(x+z)−d(x−z)|Vδ/2(z)ϕ(x)dzdx =:T21+T22. Consider $${\mathscr{T}}_2^1$$. By Lemma A.7,   |ϕ(x+z)−ϕ(x)|≤w1,ϕ(|z|)ϕ(x). Hence, by the symmetry of $$V$$ and the triangle inequality,   |T21| ≤4∬Rd×Rd|d(x−z)|Vδ/2(z)w1,ϕ(|z|)ϕ(x)dzdx ≤4w1,ϕ(δ/2)∬Rd×Rd|d(y)|Vδ/2(x−y)ϕ(x)dydx ≤2w1,ϕ(δ)‖uΔt(τ2)−uΔt(τ1)‖1,ϕ⋆Vδ/2. By Lemma A.8 and Corollary 4.3, $${E \left[{\left|{{\mathscr{T}}_2^1}\right]}\right|} = {\mathcal{O}}(\delta)$$. By Proposition 4.4, it follows in view of assumption (4.23) that $${E \left[{\left|{{\mathscr{T}}_2^2}\right]}\right|} = {\mathcal{O}}(\delta^\kappa)$$. Consequently,   T1+T2=O(δκ−1(l−k)Δt+τ2−τ1+δκ). Choosing $$\delta=((l-k){{{\it {\Delta}} t}})$$ concludes the proof of (i). Result (ii) follows analoguously due to (4.25). □ 5. Convergence Equipped with $${{{\it {\Delta}} t}}$$-uniform a priori estimates, we are now prepared to study the limiting behavior of $${u_{{{{\it {\Delta}} t}}}}, {v_{{{{\it {\Delta}} t}}}}$$ as $${{{\it {\Delta}} t}}\downarrow 0$$. As discussed in Section 1, we will apply the framework of Young measures. We refer to the appendix (Section A.5) for some background material on Young measures and weak compactness. We start by establishing an approximate entropy inequality for the operator splitting solutions. Lemma 5.1 Suppose $$u^0 \in L^2({\it {\Omega}},{\mathscr{F}}_0,P;L^2({\mathbb{R}}^d,\phi))$$, $$\phi \in \mathfrak{N}$$. Let $${u_{{{{\it {\Delta}} t}}}}$$ and $${v_{{{{\it {\Delta}} t}}}}$$ be defined by (3.4) and (3.5), respectively. For any $$(S,Q) \in \mathscr{E}$$, any $$V \in {\mathcal{S}}$$ and any non-negative $${\varphi} \in C^\infty_c([0,T) \times {\mathbb{R}}^d)$$,   0 ≤E[∫RdS(u0(x)−V)φ(0,x)dx] +E[∬ΠTS(uΔt(t,x)−V)∂tφ(t,x)+Q(vΔt(t,x),V)⋅∇φ(t,x)dxdt] −E[∬ΠTS″(uΔt(t,x)−V)DtVσ(x,uΔt(t,x))φ(t,x)dxdt] +E[12∬ΠTS″(uΔt(t,x)−V)σ2(x,uΔt(t,x))φ(t,x)dxdt] +∑n=0N−1E[∫tntn+1∫Rd(S(vΔt(t,x)−V)−S(vΔt((tn+1)−,x)−V)∂tφ(t,x)dxdt]. (5.1) Proof. Let us for the moment assume that $$u^0 \in L^p({\it {\Omega}},{\mathscr{F}}_0,P;L^p({\mathbb{R}}^d,\phi))$$ for all $$2 \leq p < \infty$$. By definition, $${v_{{{{\it {\Delta}} t}}}}$$ satisfies    ∫RdS(un(x)−V)φ(tn,x)dx−∫RdS(vΔt((tn+1)−,x)−V)φ(tn+1,x)dx +∫tntn+1∫RdS(vΔt(t,x)−V)∂tφ(t,x)dxdt +∫tntn+1∫RdQ(vΔt(t,x),V)⋅∇φ(t,x)dxdt≥0. For fixed $$x \in {\mathbb{R}}^d$$, apply Theorem A.10 with $$F(\zeta,\lambda,t) = S(\zeta-\lambda)\varphi(t,x)$$ and   uΔt(t,x)⏟X(t)=uΔt((tn)+,x)⏟X0+∫tntσ(x,uΔt(s,x))⏟u(s)dB(s). This yields, after integrating in space,   ∫RdS(un+1(x)−V)φ(tn+1,x)dx =∫RdS(uΔt((tn)+)−V)φ(tn,x)dx +∫Rd∫tntn+1S(uΔt(t,x)−V)∂tφ(t,x)dtdx +∫Rd∫tntn+1S′(uΔt(t,x)−V)σ(x,uΔt(t,x))φ(t,x)dB(t)dx −∫Rd∫tntn+1S″(uΔt(t,x)−V)DtVσ(x,uΔt(t,x))φ(t,x)dtdx +12∫Rd∫tntn+1S″(uΔt(t,x)−V)σ2(x,uΔt(t,x))φ(t,x)dtdx, where the stochastic integral is a Skorohod integral. Note that    ∫RdS(uΔt((tn)+,x)−V)φ(tn,x)dx−∫RdS(vΔt((tn+1)−,x)−V)φ(tn+1,x)dx =−∫Rd∫tntn+1S(vΔt((tn+1)−,x)−V)∂tφ(t,x)dtdx. Adding the two equations and taking expectations we attain   E[∫RdS(un(x)−V)φ(tn,x)dx−E[∫RdS(un+1(x)−V)φ(tn+1,x)dx]] +E[∫tntn+1∫Rd(S(vΔt(t,x)−V)−S(vΔt((tn+1)−,x)−V))∂tφ(t,x)dxdt] +E[∫Rd∫tntn+1S(uΔt(t,x)−V)∂tφ(t,x)dtdx] +E[∫tntn+1∫RdQ(vΔt(t,x),V)⋅∇φ(t,x)dxdt] −E[∫Rd∫tntn+1S″(uΔt(t,x)−V)DtVσ(x,uΔt(t,x))φ(t,x)dtdx] +E[12∫Rd∫tntn+1S″(uΔt(t,x)−V)σ2(x,uΔt(t,x))φ(t,x)dtdx]≥0, where we applied the fact that the Skorohod integral has zero expectation. Next we sum over $$n = 0,1,\dots,N-1$$. This yields (5.1). The result follows for general $$u^0 \in L^2({\it {\Omega}},{\mathscr{F}}_0,P;L^2({\mathbb{R}}^d,\phi))$$ by approximation. □ Theorem 5.2 Suppose (𝓐f), (𝓐f,1), (𝓐σ) and (𝓐σ,1) hold. Let $$\phi \in \mathfrak{N}$$ and $$2 \le p < \infty$$. Suppose $$u^0 \in L^p({\it {\Omega}},{\mathscr{F}}_0,P;L^p({\mathbb{R}}^d,\phi))$$ satisfies (4.23). Let $${u_{{{{\it {\Delta}} t}}}}$$ and $${v_{{{{\it {\Delta}} t}}}}$$ be defined by (3.4) and (3.5), respectively. Then there exists a subsequence $$\left\{{{{{\it {\Delta}} t}}_j}\right\}$$ and a predictable $$u \in L^p([0,T] \times {\it {\Omega}};L^p({\mathbb{R}}^d \times [0,1],\phi))$$ such that both $${u_{{{{\it {\Delta}} t}}}}j \rightarrow u$$ and $${v_{{{{\it {\Delta}} t}}}}j \rightarrow u$$ in the following sense: for any Carathéodory function $$\Psi:{\mathbb{R}} \times {\it {\Pi}}_T \times {\it {\Omega}} \rightarrow {\mathbb{R}}$$ such that $$\Psi({u_{{{{\it {\Delta}} t}}}}j,\cdot) \rightharpoonup \overline{\Psi}$$ (respectively, $$\Psi({v_{{{{\it {\Delta}} t}}}}j,\cdot) \rightharpoonup \overline{\Psi}$$) in $$L^1({\it {\Pi}}_T \times {\it {\Omega}},\phi\,\mathrm{d}x \otimes \mathrm{d}t \otimes {\rm d}P)$$,   Ψ¯(t,x,ω)=∫01Ψ(u(t,x,α,ω),t,x,ω)dα. (5.2) The process $$\tilde{u} = \int_0^1 u \,{\rm d}\alpha$$ is an entropy solution in the sense of Definition 2.1 with initial condition $$u^0$$. Proof. 1. Existence of limits. Let us investigate the limit behavior of $${u_{{{{\it {\Delta}} t}}}}$$, noting that the same considerations apply to $${v_{{{{\it {\Delta}} t}}}}$$. We argue as in Karlsen & Storrøsten (2017, Theorem 4.1, Step 1) (see also (see also Bauzet et al., 2012, Section A.3.3). We apply Theorem A.11 to $$\left\{{{u_{{{{\it {\Delta}} t}}}}}\right\}$$ on the measure space   (X,A,μ)=(Ω×ΠT,P⊗B(Rd),dP⊗dt⊗ϕdx). By Corollary 4.3,   supΔt>0{E[∬ΠT|uΔt|2ϕ(x)dxdt]}<∞. Hence, there exists a Young measure $$\nu = \nu_{t,x,\omega}$$ such that for any Carathéodory function $$\Psi$$ satisfying $$\Psi({u_{{{{\it {\Delta}} t}}}}j,\cdot) \rightharpoonup \overline{\Psi}$$ in $$L^1({\it {\Pi}}_T \times {\it {\Omega}},\phi\,\mathrm{d}x \otimes \mathrm{d}t \otimes {\rm d}P)$$, it follows that   Ψ¯(t,x,ω)=∫RΨ(ξ,t,x,ω)dνt,x,ω(ξ). Define (Panov, 1996; Eymard et al., 2000)   u(t,x,α,ω):=inf{ξ∈R:νt,x,ω((−∞,ξ])>α}. The representation (5.2) follows from the relation $${\mathcal{L}} \circ u^{-1}(t,x,\cdot,\omega) = \nu_{t,x,\omega}$$, where $${\mathcal{L}}$$ denotes the Lebesgue measure on $$[0,1]$$. For predictability and the fact that $$u \in L^p([0,T] \times {\it {\Omega}};L^p({\mathbb{R}}^d \times [0,1],\phi))$$ see Karlsen & Storrøsten (2017, Theorem 4.1); Panov (1996, Section 3); Bauzet et al. (2012, Section A.3.3). 2. Independence of interpolation. Denote by $$v$$ the limit of $$\left\{{{v_{{{{\it {\Delta}} t}}}}}\right\}$$; see step 1. We want to show that $$v = u$$. By Pedregal (1997, Lemma 6.3), this holds true if   T(Δt):=E[∬ΠT|uΔt(t,x)−vΔt(t,x)|ϕ(x)dtdx]→0 as Δt↓0. (5.3) To see this, observe that   T ≤E[∑n=0N−1∫tntn+1∫Rd|uΔt(t,x)−uΔt((tn)+,x)|ϕ(x)dtdx] +E[∑n=0N−1∫tntn+1∫Rd|vΔt((tn+1)−,x)−vΔt(t,x)|ϕ(x)dtdx] =:T1+T2. By Proposition 4.8 (i),   T1≤CT,ϕ∑n=0N−1∫tntn+1t−tndt=23CT,ϕTΔt. By Proposition 4.8 (ii),   T2≤CT,ϕ∑n=0N−1∫tntn+1(tn+1−t)κdt≤CT,ϕTΔtκ, where $$\kappa$$ is defined in Proposition 4.8. This proves (5.3). 3. Entropy inequality. We need to prove that $$u$$ is a Young measure-valued entropy solution in the sense of Karlsen & Storrøsten (2017, Definition 2.2). The result then follows from Karlsen & Storrøsten (2017, Theorem 5.1). Let $$S,V,{\varphi}$$ be as in Lemma 5.1 and define   TΔt:=∑n=0N−1E[∫tntn+1∫Rdbig(S(vΔt(t,x)−V)−S(vΔt((tn+1)−,x)−V))∂tφdxdt]. We want to show that $${\mathscr{T}}_{{{\it {\Delta}} t}} \rightarrow 0$$ as $${{{\it {\Delta}} t}} \downarrow 0$$. Recall the definition of the weighted $$L^\infty$$-norm (2.1). By Proposition 4.8,   |TΔt| ≤‖S‖Lipsupt∈[0,T]{‖∂tφ‖∞,ϕ−1} ×∑n=0N−1E[∫tntn+1∫Rd|vΔt(t,x)−vΔt((tn+1)−,x)|ϕ(x)dxdt] ≤‖S‖Lipsupt∈[0,T]{‖∂tφ‖∞,ϕ−1}CT,ϕTΔtκ, as in the proof of step 2. Concerning the remaining terms in Lemma 5.1, the limit $${{{\it {\Delta}} t}} \downarrow 0$$ is treated exactly as in Karlsen & Storrøsten (2017, Proof of Theorem 4.1, Step 2). It follows that $$u$$ is a Young measure-valued entropy solution. □ 6. Error estimate We now restrict our attention to the case   σ(x,u)=σ(u),σ∈L∞. (𝓐σ,2 As mentioned in Section 1, for homogeneous noise functions $$\sigma = \sigma(u)$$, whenever $${E \left[{\left\Vert{\nabla u_0}\right\Vert_{1,\phi}}\right]} < \infty$$, the entropy solution $$u$$ to (1.1) satisfies a spatial BV estimate of the form   E[∫Rd|∇u(t,x)|ϕ(x)dx]≤C(0≤t≤T) (6.1) for some finite constant $$C$$ (depending on $$u_0,f,\phi,\sigma,T$$). Here $$\nabla u(t,\cdot)$$ is a (locally finite) measure and $$\phi \in \mathfrak{N}$$. This can be seen as a consequence of the fractional space translation estimate (2.3) and Remark 4.7. A direct verification of (6.1) can also be found in Chen et al. (2012, Theorem 2.1) (when $$\phi \equiv 1$$). The same estimate is available for the operator splitting solution; cf. Proposition 4.4. For the error estimate, we consider yet another time interpolation $$\eta_{{{\it {\Delta}} t}}$$ of the operator splitting $$\left\{{u^n}\right\}_{n=0}^N$$. Inspired by Langseth et al. (1996), let   ηΔt(t):=(SSDE(t,tn)−I)SCL(Δt)un⏟uΔt(t)−SCL(Δt)un+SCL(t−tn)un,⏟vΔt(t)t∈[tn,tn+1]. (6.2) A graphical representation of the interpolation $$\eta_{{{\it {\Delta}} t}}$$ is given in Fig. 2. Fig. 2. View largeDownload slide A graphical representation of $$\eta_{{{\it {\Delta}} t}}$$. The value of $$\eta_{{{\it {\Delta}} t}}(t)$$ corresponds to summing (with signs) the values taken at the unfilled dots. Fig. 2. View largeDownload slide A graphical representation of $$\eta_{{{\it {\Delta}} t}}$$. The value of $$\eta_{{{\it {\Delta}} t}}(t)$$ corresponds to summing (with signs) the values taken at the unfilled dots. Theorem 6.1 Fix $$\phi \in \mathfrak{N}$$. Suppose (𝓐f), (𝓐f,1), (𝓐σ) and (𝓐σ,2) are satisfied. Suppose also that $$u^0,u_0 \in L^2({\it {\Omega}},{\mathscr{F}}_0,P;L^2({\mathbb{R}}^d,\phi))$$ satisfy (4.23) with $${{\kappa_0}} = 1$$. Let $$u$$ be the entropy solution of (1.1) and (1.2) according to Definition 2.1 with initial condition $$u_0$$, and let $$\eta_{{{\it {\Delta}} t}}$$ be defined by (6.2). Then there exists a constant $$C$$, independent of $${{{\it {\Delta}} t}}$$ but dependent on $$\sigma,f,T,\phi,u_0,u^0$$, such that   E[‖u(t)−ηΔt(t)‖1,ϕ]≤eCϕ‖f‖Lipt(E[‖u0−u0‖1,ϕ]+CΔt1/3),t∈[0,T]. The proof is split into several parts, the results of which are gathered toward the end of the section. To help motivate the upcoming technical arguments, let us outline a ‘high-level’ overview of the main idea, assuming that all relevant functions are smooth in $$x$$ and the spatial dimension is $$d = 1$$. The function $$\eta_{{{\it {\Delta}} t}}$$ defined in (6.2) ought to satisfy an ‘approximate’ entropy inequality. Formally, we have   dηΔt+∂xf(vΔt)dt=σ(uΔt)dB, (6.3) indicating that the error terms can be expressed as perturbations of the coefficients $$f,\sigma$$. Let $$u$$ be a smooth (in $$x$$) solution of (1.1). By (6.3),   d(ηΔt−u)=−∂x(f(vΔt)−f(u))dt+(σ(uΔt)−σ(u))dB, and thus the Itô formula gives   dS(ηΔt−u) =−S′(ηΔt−u)∂x(f(vΔt)−f(u))dt+S′(ηΔt−u)(σ(uΔt)−σ(u))dB +12S″(ηΔt−u)(σ(uΔt)−σ(u))2dt, for any $$S \in C^2({\mathbb{R}})$$. Upon adding and subtracting identical terms and taking expectations, we arrive at   E[dS(ηΔt−u)]= −E[S′(ηΔt−u)∂x(f(ηΔt)−f(u))dt] +12E[S″(ηΔt−u)(σ(ηΔt)−σ(u))2dt] +E[S′(ηΔt−u)∂x(f(ηΔt)−f(vΔt))dt] +E[S″(ηΔt−u)(∫ηΔtuΔt(σ(z)−σ(u))σ′(z)dz)dt]. The first two terms vanish as $$S\rightarrow \left|{\cdot}\right|$$. Note that these terms also appear in the uniqueness argument, when two exact solutions are compared. Accordingly, they should not be thought of as error terms originating from the splitting procedure. The last two terms, however, are genuine error terms associated with the operator splitting and the interpolation $$\eta_{{{\it {\Delta}} t}}$$. All of the above terms may be recognized in the forthcoming Lemma 6.3. The above simplified representation provides intuition on how to estimate these error terms. This is, in particular, the case concerning the third term on the right-hand side. To this end, note that   ηΔt−vΔt=uΔt−SCL(Δt)un=∫tntσ(uΔt(s))dB(s) for $$t_n \leq t < t_{n+1}$$. Consequently,   ∂x(f(ηΔt)−f(vΔt)) =(f′(ηΔt)−f′(vΔt))∂xvΔt +f′(ηΔt)∫tnt∂xσ(uΔt(s))dB(s). (6.4) Furthermore,   E[|(f′(ηΔt)−f′(vΔt))∂xvΔt|]≤‖f′‖LipE[E[|∫tntσ(uΔt(s))dB(s)||Ftn]|∂xvΔt|], which provides a way to estimate the term since $${v_{{{{\it {\Delta}} t}}}}(t)\in BV$$ and $$\sigma\in L^\infty$$. Because of the lack of regularity, we will work with an approximation of $$\eta_{{{\it {\Delta}} t}}$$. Given $$\left\{{w^n = w^n(x)}\right\}_{n = 0}^{N-1}$$, we set   ψ(t):=(SSDE(t,tn)−I)wn,t∈[tn,tn+1), (6.5) and $${{\tilde{\eta}}} := \psi + {v_{{{{\it {\Delta}} t}}}}$$. Note that $${\eta_{{{\it {\Delta}} t}}} = \psi + {v_{{{{\it {\Delta}} t}}}}$$ whenever $$w^n = {\mathcal{S}_{\text{CL}}}({{{\it {\Delta}} t}})u^n$$ for $$n = 0,\dots,N-1$$. However, because of the lack of differentiability of $${\mathcal{S}_{\text{CL}}}({{{\it {\Delta}} t}})u^n$$, we will work with a sequence $$\left\{{w^n_k}\right\}_{k \geq 1}$$ of smooth functions satisfying $$w^n_k \rightarrow {\mathcal{S}_{\text{CL}}}({{{\it {\Delta}} t}})u^n$$ in $$L^1({\it {\Omega}};L^1({\mathbb{R}}^d,\phi))$$ as $$k \rightarrow \infty$$. To simplify notation, we suppress the dependence on $$k$$ and write $$w^n = w^n_k$$. Proposition 6.2 Suppose (𝓐f), (𝓐σ) and (𝓐σ,2) are satisfied. Let $${{\tilde{\eta}}} = \psi + {v_{{{{\it {\Delta}} t}}}}$$, where $$\psi$$ and $${v_{{{{\it {\Delta}} t}}}}$$ are defined in (6.5) and (3.5), respectively. Then, for all non-negative $$\phi \in C^\infty_c([t_n,t_{n+1}]\times {\mathbb{R}}^d)$$, any $$V \in {\mathbb{D}}^{1,2}$$, and all entropy/entropy–flux pairs $$(S,Q) \in \mathscr{E}$$,   E[∫RdS(η~(tn,x)−V)ϕ(tn,x)dx] −E[∫RdS(η~((tn+1)−,x)−V)ϕ(tn+1,x)dx] +E[∬ΠnS(η~−V)∂tϕ+Q(η~,V)⋅∇ϕdxdt] +E[∬Πn∫Vη~S′(z−V)(f′(z−ψ)−f′(z))dz⋅∇ϕdxdt] +E[∬Πn∫Vη~S″(z−V)f′(z−ψ)dz⋅∇ψϕdxdt] −E[∬ΠnS″(η~−V)DtVσ(ψ+wn)ϕdxdt] +12E[∬ΠnS″(η~−V)σ2(ψ+wn)ϕdxdt]≥0, where $${\it {\Pi}}_n = [t_n,t_{n+1}] \times {\mathbb{R}}$$. The proof of Proposition 6.2 is deferred to Section A.1. To ensure that the relevant quantities are Malliavin differentiable, we replace the entropy solution $$u$$ by the viscous approximation $${{u^\varepsilon}}$$, which solves   duε+∇⋅f(uε)dt=σ(x,uε)dB(t)+εΔuεdt,uε(0)=u0, and then send $$\varepsilon \downarrow 0$$ at a later stage. Let us recall that $$\left\{{D_r{{u^\varepsilon}}(t)}\right\}_{t > r}$$ is a predictable weak solution to the linear problem   dw+∇⋅(f′(uε)w)dt=σ′(x,uε)wdB(t)+εΔwdt,w(r)=σ(uε(r)), for almost all $$r \in [0,T]$$ (cf. Karlsen & StorrØsten, 2017, Section 3). Furthermore,   esssupr∈[0,T]supt∈[0,T]{E [ ∥Druε(t)∥2,ϕ2}<∞. As a consequence of Karlsen & StorrØsten (2017, Theorem 5.1) and Pedregal (1997, Proposition 6.12), we have $${{u^\varepsilon}} \rightarrow u$$ in $$L^1([0,T] \times {\it {\Omega}};L^1({\mathbb{R}}^d,\phi))$$ as $$\varepsilon \downarrow 0$$. In fact, under the assumptions of Theorem 6.1, $${{u^\varepsilon}} \rightarrow u$$ with rate $$1/2$$ (Chen et al., 2012, Theorem 5.2). We may now proceed with the doubling-of-the-variables argument. Lemma 6.3 Fix $$\phi \in \mathfrak{N}$$. Let $${{u^\varepsilon}} = {{u^\varepsilon}}(s,y)$$ be the viscous approximation of (1.1). Take $$w(t,x) = w^n(x)$$ for $$t \in [t_n,t_{n+1})$$, and let $$\psi = \psi(t,x)$$, $${v_{{{{\it {\Delta}} t}}}} = {v_{{{{\it {\Delta}} t}}}}(t,x)$$ and $${{\tilde{\eta}}} = {{\tilde{\eta}}}(t,x)$$ be defined in Proposition 6.2. Let $$t_0 \in [0,T)$$, and pick $${\it {\gamma}},r_0,r > 0$$ such that $$t_0 \leq T-2({\it {\gamma}} + r_0)$$. Define   ξγ(t)=1−∫0tJγ+(s−t0)ds. Furthermore, let   φ(t,x,s,y)=12dϕ(x+y2)Jr(x−y2)Jr0+(s−t)ξγ(t), and $$S_\delta$$ be defined in (4.9). Then   L−R+F+T1+T2+T3+T4+T5+T6≥0, (6.6) where   L =E[∬ΠT∫RdSδ(η~(0,x)−uε(s,y))φ(0,x,s,y)dxdsdy],R =−E[⨌ΠT2Sδ(η~−uε)(∂t+∂s)φdX],F =E[⨌ΠT2Q(uε,η~)⋅∇yφ+Q(η~,uε)⋅∇xφdX],T1 =12E[⨌ΠT2Sδ″(uε−η~)(σ(uε)−σ(η~))2φdX],T2 =E[⨌ΠT2Sδ″(uε−η~)(σ(uε)−Dtuε)σ(ψ+w)φdX],T3 =E[⨌ΠT2Sδ″(uε−η~)(∫η~ψ+w(σ(z)−σ(uε))σ′(z)dz)φdX],T4 =E[⨌ΠT2∫uεη~Sδ′(z−uε)(f′(z−ψ)−f′(z))dz⋅∇xφdX] +E[⨌ΠT2∫uεη~Sδ″(z−uε)f′(z−ψ)dz⋅∇xψφdX],T5 =εE[⨌ΠT2Sδ(uε−η~)ΔyφdX],T6 =∑n=0N−1E[∬ΠT∫Rd(Sδ(η~((tn+1),x)−uε(s,y)) −Sδ(η~((tn+1)−,x)−uε(s,y)))φ(tn+1,x,s,y)dxdsdy], where $$\mathrm{d}X = \mathrm{d}x\,\mathrm{d}t\,\mathrm{d}s\,\mathrm{d}y$$. Proof. Let us first assume $$\phi \in C^\infty_c({\mathbb{R}}^d)$$, as the result for $$\phi\in \mathfrak{N}$$ then follows from an approximation argument. After a standard application of Itô’s formula to $${{u^\varepsilon}}(s,y) \mapsto S_\delta({{u^\varepsilon}}(s,y)-{{\tilde{\eta}}}(t,x)) \varphi(s)$$ for $$s \geq t$$, we arrive at   E[⨌ΠT2Sδ(uε−η~)∂sφ+Q(uε,η~)⋅∇yφdX] +12E[⨌ΠT2Sδ″(uε−η~)σ2(uε)φdX]+εE[⨌ΠT2Sδ(uε−η~)ΔyφdX]≥0; cf. Karlsen & Storrøsten (2017, Lemma 5.3). Take $$V = {{u^\varepsilon}}(s,y)$$ in Proposition 6.2, integrate in $$(s,y) \in {\it {\Pi}}_T$$ and sum over $$n = 0,\dots,N-1$$. The outcome is   E[∬ΠT∫RdSδ(η~(0,x)−uε(s,y))φ(0,x,s,y)dxdsdy] +E[⨌ΠT2Sδ(η~−uε)∂tφ+Q(η~,uε)⋅∇xφdX] +E[⨌ΠT2∫uεη~Sδ′(z−uε)(f′(z−ψ)−f′(z))dz⋅∇xφdX] +E[⨌ΠT2∫uεη~Sδ″(z−uε)f′(z−ψ)dz⋅∇xψφdX] −E[⨌ΠT2Sδ″(η~−uε)Dtuεσ(ψ+w)φdX] +12E[⨌ΠT2Sδ″(η~−uε)σ2(ψ+w)φdX] +∑n=0N−1E[∬ΠT∫Rd(Sδ(η~((tn+1),x)−uε(s,y)) −Sδ(η~((tn+1)−,x)−uε(s,y)))φ(tn+1,x,s,y)dxdsdy]≥0. The lemma follows upon adding the two previous inequalities, noting that   12σ2(uε)−Dtuεσ(ψ+w)+12σ2(ψ+w) =12(σ(ψ+w)−σ(uε))2+(σ(uε)−Dtuε)σ(ψ+w) =12(σ(η~)−σ(uε))2+∫η~ψ+w(σ(z)−σ(uε))σ′(z)dz +(σ(uε)−Dtuε)σ(ψ+w). □ In the following we estimate the terms appearing in Lemma 6.3. The underlying assumptions are the ones made in Theorem 6.1. We let $$C$$ denote a generic constant, meaning that it is independent of the ‘small’ parameters $${{{\it {\Delta}} t}},r,r_0,{\it {\gamma}},\varepsilon,\delta$$. Furthermore, given a term $${\mathscr{T}}$$, we write $${\mathscr{T}} = \mathcal{O}(g({{{\it {\Delta}} t}}, \dots,\delta))$$ whenever $$\left|{{\mathscr{T}}}\right| \leq Cg({{{\it {\Delta}} t}}, \dots,\delta)$$ for some non-negative function $$g$$. Estimate 6.4 Let $$L$$ be defined in Lemma 6.3. Then   lim supr0↓0L≤E[‖u0−u0‖1,ϕ]+O(δ+r). Proof. By (4.10),   |Sδ(η~(0,x)−uε(s,y))−|η~(0,x)−uε(s,y)||≤δ. By the reverse triangle inequality,   ||η~(0,x)−uε(s,y)|−|η~(0,x)−u0(y)|| ≤|uε(s,y)−u0(y)|,||η~(0,x)−u0(y)|−|η~(0,x)−u0(x)|| ≤|u0(y)−u0(x)|. Hence, after adding and subtracting identical terms, noting that $${{\tilde{\eta}}}(0) = u^0$$, it follows by the triangle inequality that   |Sδ(η~(0,x)−uε(s,y))−|u0(x)−u0(x)||≤δ+|uε(s,y)−u0(y)|+|u0(y)−u0(x)|. By (4.15),   |L−E[‖u0−u0‖1,ϕ⋆Jr]| ≤δ‖ϕ‖L1(Rd)+∫0TE[‖uε(s)−u0‖1,ϕ⋆Jr]Jr0+(s)ds⏟Z1 +E[12d∬Rd×Rd|u0(y)−u0(x)|ϕ(x+y2)Jr(x−y2)dxdy]Z2⏟. Thanks to Karlsen & Storrøsten (2017, Lemma 2.3), $${\mathscr{Z}}_1 \to 0$$ as $$r_0 \to 0$$. Regarding $${\mathscr{Z}}_2$$ we apply (4.14). As $$u_0$$ satisfies (4.23) with $${{\kappa_0}} = 1$$,   Z2=E[∬Rd×Rd|u0(x+z)−u0(x−z)|ϕ(x)Jr(z)dxdz]=O(r). Finally, we apply Lemma A.8 to conclude that   |E[‖u0−u0‖1,ϕ⋆Jr−‖u0−u0‖1,ϕ]|=O(r). □ Estimate 6.5 Let $$R$$ be defined in Lemma 6.3. Then   lim infε,r0↓0R≥E[∫0T‖η~(t)−u(t)‖1,ϕJγ+(t−t0)dt]+O(δ+r). Proof. It is easy to check that   R=E[⨌ΠT2Sδ(η~(t,x)−uε(s,y))12dϕ(x+y2)×Jr(x−y2)Jr0+(s−t)Jγ+(t−t0)dX]. Moreover, adding and subtracting identical terms, we obtain   |Sδ(η~(t,x)−uε(s,y))−|η~(t,x)−uε(t,x)||≤δ+|uε(s,y)−uε(t,y)|+|uε(t,y)−uε(t,x)|, and so   |R−E[∫0T‖η~(t)−uε(t)‖1,ϕ⋆JrJγ+(t−t0)dt]| ≤δ‖ϕ‖L1(Rd)+E[∬[0,T]2‖uε(s)−uε(t)‖ϕ⋆JrJr0+(s−t)Jγ+(t−t0)dsdt]⏟Z1 +E[∬ΠT∫Rd|uε(t,y)−uε(t,x)|12dϕ(x+y2)Jr(x−y2)Jγ+(t−t0)dxdydt]⏟Z2. Because of Lemma A.9, $$\lim_{r_0 \downarrow 0} {\mathscr{Z}}_1 = 0$$. Next, we utilize the strong convergence $${{u^\varepsilon}} \rightarrow u$$ in $$L^1([0,T] \times {\it {\Omega}};L^1({\mathbb{R}}^d,\phi))$$ and (4.14) to conclude that   limε,r0↓0Z2=∫0TE[∬Rd×Rd|u(t,x+z)−u(t,x−z)|ϕ(x)Jr(z)dxdz]Jγ+(t−t0)dt. It follows from Karlsen & Storrøsten (2017, Proposition 5.2) and the assumption (4.23) with $${{\kappa_0}} = 1$$ that $$\left|{\lim_{\varepsilon,r_0 \downarrow 0} {\mathscr{Z}}_2}\right| = \mathcal{O}(r)$$. The claim is now a consequence of Lemma A.8. □ Estimate 6.6 Let $$F$$ be defined in Lemma 6.3. Then   lim supε,r0↓0F≤Cϕ‖f‖LipE[∫0T‖u(t)−η~(t)‖1,ϕξγ(t)dt]+O(δ(1+1r)+r). Proof. Observe that   F=F1+F2+F3, (6.7) where   F1 :=E[⨌ΠT2Sδ′(uε−η~)(f(uε)−f(η~))(∇x+∇y)φdX],F2 :=−E[⨌ΠT2∫uεη~Sδ″(z−uε)(f(z)−f(uε))dz⋅∇xφdX],F3 :=−E[⨌ΠT2∫η~uεSδ″(z−η~)(f(z)−f(η~))dz⋅∇yφdX]. The decomposition (6.7) follows from the identities   Qδ(uε,η~) =Sδ′(uε−η~)(f(uε)−f(η~))−∫η~uεSδ″(z−η~)(f(z)−f(η~))dz,Qδ(η~,uε) =Sδ′(η~−uε)(f(η~)−f(uε))−∫uεη~Sδ″(z−uε)(f(z)−f(uε))dz, derived using integration by parts. Next, we claim that   |F2|+|F3|=O(δ(1+1r)). (6.8) We consider $$F_2$$; the $$F_3$$ term is estimated likewise. Note that   |∫uεη~Sδ″(z−uε)(f(z)−f(uε))dz|≤‖f‖Lipδ. Hence,   |F2|≤‖f‖LipδE[⨌ΠT2|∇xφ|dX]. By a straightforward computation,   ⨌ΠT2|∇xφ|dX≤12T(Cϕ+‖∇J‖L1(Rd)1r)‖ϕ‖L1(Rd). This proves (6.8). Next, we claim that   lim supε,r0↓0F1≤Cϕ‖f‖LipE[∫0T‖u(t)−η~(t)‖1,ϕ∗Jrξγ(t)dt]+O(δ+r). (6.9) Set   Fδ(b,a)=Sδ′(b−a)(f(b)−f(a)). Then   |Fδ(b,a)−Fδ(c,a)| =|∫cb∂z(Sδ′(z−a)(f(z)−f(a)))dz| ≤2‖f‖Lipδ+‖f‖Lip|b−c|; whence   |Fδ(uε(s,y),η~(t,x))−Fδ(uε(t,x),η~(t,x))|≤‖f‖Lip(2δ+|uε(s,y)−uε(t,y)|+|uε(t,y)−uε(t,x)|), and so   F1−E∬ΠTFδ(uε(t,x),η~(t,x))⋅(∇ϕ∗Jr)(x)ξγ(t)dxdt ≤Cϕ‖f‖LipE[∬[0,T]2‖uε(s)−uε(t)‖1,ϕ⋆JrJr0+(s−t)ξγ(t)dsdt]  +Cϕ‖f‖LipE[∫0T∬Rd×Rd|uε(t,x+z)−uε(t,x−z)|Jr(z)ξγ(t)ϕ(x)dxdzdt]  +2δ‖f‖LipT‖∇ϕ‖L1(Rd), where we have made a change of variables as in Estimate 6.5. Following the same reasoning as in that estimate we arrive at   lim supε,r0↓0F1≤E[∬ΠTFδ(u(t,x),η~(t,x))⋅(∇ϕ∗Jr)(x)ξγ(t)dxdt]+O(δ+r). Inequality (6.9) follows from $$\mathcal{F}_\delta(a,b) \le \left\Vert{f}\right\Vert_\mathrm{Lip}\left|{a-b}\right|$$ and $$\left|{\nabla \phi}\right| \le C_\phi\phi$$. Combining the above estimates for $$F_1,F_2,F_3$$ concludes the proof of the claim. □ Estimate 6.7 Let $${\mathscr{T}}_1$$ be defined in Lemma 6.3. Then   |T1|≤Cδ. Proof. Since $$S_\delta'' = 2J_\delta$$,   Sδ″(uε−η~)(σ(uε)−σ(η~))2≤2‖σ‖Lip2Jδ(uε−η~)|uε−η~|2≤2‖σ‖Lip2‖J‖∞δ. Because of (4.15) and Young’s inequality for convolutions,   ⨌ΠT2φdX=(∫0T∫0TJr0+(s−t)ξγ(t)dsdt)(∫Rdϕ⋆Jr(x)dx)≤T‖ϕ‖L1(Rd). The result follows. □ Estimate 6.8 Let $${\mathscr{T}}_2$$ be defined in Lemma 6.3. Then   limr0↓0T2=0. Proof. This follows exactly as in Karlsen & StorrØsten (2017, Limit 5). However, the assumption $$\sigma \in L^\infty$$ simplifies the analysis and allows for $$\phi \in \mathfrak{N}$$ instead of $$C^\infty_c({\mathbb{R}}^d)$$. □ Estimate 6.9 Let $${\mathscr{T}}_3$$ be defined in Lemma 6.3. Then   |T3|≤C1δE[∑n=0N−1∫tntn+1‖wn−vΔt(t)‖ϕ⋆Jrdt]. Proof. Now, as $${{\tilde{\eta}}} = \psi + {v_{{{{\it {\Delta}} t}}}}$$,   |∫η~ψ+w(σ(z)−σ(uε))σ′(z)dz|≤2‖σ‖∞‖σ‖Lip|w−vΔt|. Keep in mind that $$w(t) = w^n$$ for $$t \in [t_n,t_{n+1})$$. The estimate then follows from (4.10) and (4.15). □ Estimate 6.10 Let $${\mathscr{T}}_4$$ be defined in Lemma 6.3. Then   |T4|≤CΔt(1+E[∫0T‖∇w(t)‖1,ϕ⋆Jr]). Proof. The estimate is established under the assumption that $${v_{{{{\it {\Delta}} t}}}}$$ is smooth in $$x$$. The general result follows by an approximation argument. Integrating by parts and using the chain rule,   T4 =E[⨌ΠT2∫uεη~Sδ′(z−uε)(f′(z−ψ)−f′(z))dz⋅∇xφdX] =−E[⨌ΠT2Sδ′(η~−uε)(f′(vΔt)−f′(η~))⋅∇xη~φdX] +E[⨌ΠT2∫uεη~Sδ′(z−uε)f″(z−ψ)dz⋅∇xψφdX]. Next, we observe that   ∫uεη~Sδ′(z−uε)f″(z−ψ)dz=−∫uεη~Sδ″(z−uε)f′(z−ψ)dz+Sδ′(η~−uε)f′(vΔt). Therefore,   T4=E[∫∫∫∫ΠT2⁡Sδ′(η~−uε)f′(η~)⋅∇xψφdX]⏟Z1  +E∫∫∫∫ΠT2⁡Sδ′(η~−uε)(f′(η~)−f′(vΔt))⋅∇xvΔtφdX⏟Z2; cf. (6.4). Consider $${\mathscr{Z}}_2$$. Since $${v_{{{{\it {\Delta}} t}}}}(t)$$ is $${\mathscr{F}}_{t_n}$$-measurable for all $$t \in [t_{n},t_{n+1})$$,   |Z2|≤E[∫∫∫∫ΠT2⁡|f′(η~)−f′(vΔt)||∇xvΔt|φdX] ≤‖f′‖Lip∑n=0N−1∫∫∫∫ΠT×Πn[EE[|ψ||Ftnn]|∇xvΔt|]φdX. By definition,   ψ(t,x)=∫tntσ(ψ(r,x)+wn(x))dB(r),tn≤t<tn+1. (6.10) In view of Jensen’s inequality for conditional expectation and the conditional Itô isometry (Capiński et al., 2012, Theorem 3.20),   E[|ψ(t,x)||Ftn] ≤E[|ψ(t,x)|2|Ftn]1/2 =E[∫tntσ2(ψ(t,x)+wn(x))ds|Ftn]1/2 ≤‖σ‖∞t−tn. It follows from Proposition 4.4 that   |Z2|≤‖σ‖∞‖f′‖LipΔtE [∫0T‖∇xvΔt(t)‖1,ϕ⋆Jrdt]≤CΔt. Consider $${\mathscr{Z}}_1$$. In view of (4.15),   |Z1|≤‖f‖LipE[⨌ΠT2|∇xψ|φdX]≤‖f‖LipE[∬ΠT|∇xψ|(ϕ⋆Jr)dxdt]. Differentiating (6.10) yields, for $$t_n \le t < t_{n+1}$$,   ∇xψ(t,x)=∫tntσ′(ψ(r,x)+wn(x))(∇xψ(r,x)+∇xwn(x))dB(r). By Lemma 6.11 below there is a constant $$C > 0$$, depending only on $$\sigma$$, such that   E[|∇xψ(t,x)|]≤Ct−tnE[|∇wn(x)|],tn≤t<tn+1. We conclude that   |Z1|≤C(E[∫0T‖∇w(t)‖1,ϕ∗Jr]dt)Δt. □ Lemma 6.11 Suppose $$h:[t_n,t_{n+1}] \times {\it {\Omega}} \rightarrow {\mathbb{R}}^d$$ is predictable and   P[∫tnt|h(s)|2ds<∞]=1. Suppose $$X(t_n) \in L^p({\it {\Omega}},{\mathscr{F}}_{t_n},P;{\mathbb{R}}^d)$$, $$1 \le p < \infty$$ and let $$X:[t_n,t_{n+1}] \times {\it {\Omega}} \rightarrow {\mathbb{R}}^d$$ satisfy   X(t)=X(tn)+∫tnth(s)dB(s),t∈[tn,tn+1]. Suppose there exist a constant $$K$$ and $$Y \in L^p({\it {\Omega}},{\mathscr{F}}_{t_n},P)$$ such that   |h(t;ω)|≤Y(ω)+K|X(t)|,t∈[tn,tn+1]. (6.11) Then, for all $$t \in [t_n,t_{n+1}]$$ and $$\beta > p(c_p^{1/p}K)^2/2$$,   suptn≤s≤tE[|X(s)|p]1/p≤C(β)eβ(t−tn)(E[|X(tn)|p]1/p+cp1/pt−tnE[|Y|p]1/p), where $$C(\beta) = \left(1-c_p^{1/p}K\sqrt{p/2\beta}\right)^{-1}$$ and $$c_p$$ is the constant from the Burkholder–Davis–Gundy inequality. Proof. Set   ‖X‖β,p,τ:=(suptn≤t≤τe−β(t−tn)E[|X(t)|p])1/p. The triangle inequality yields   E[|X(t)|p]1/p≤E[|∫tnth(s)dB(s)]|p1/p+E[|X(tn)|p]1/p. By the Burkholder–Davies–Gundy inequality,   E[|∫tnth(s)dB(s)|p]1/p≤cp1/pE[(∫tnth2(s)ds)p/2]1/p. Because of (6.11) and the triangle inequality on $$L^p({\it {\Omega}};L^2([t_n,t]))$$,   E[(∫tnt|h(s)|2ds)p/2]1/p≤t−tnE[|Y|p]1/p+KE[(∫tnt|X(s)|2ds)p/2]1/p. By Minkowski’s integral inequality,   E[(∫tnt|X(s)|2ds)p/2]2/p≤∫tntE[|X(s)|p]2/pds. Furthermore,   ∫tntE[|X(s)|p]2/pds =e2β(t−tn)/p∫tnt(e−β(t−s)e−β(s−tn)E[|X(s)|p])2/pds ≤e2β(t−tn)/p‖X‖β,p,t2∫tnte−2β(t−s)/pds =p2β(e2β(t−tn)/p−1)‖X‖β,p,t2. Summarizing, we arrive at   E[|X(t)|p]1/p ≤E[|X(tn)|p]1/p+cp1/pt−tnE[|Y|p]1/p +cp1/pKp2β(e2β(t−tn)/p−1)1/2‖X‖β,p,t. Multiplying by $$\,{\it e}^{-\beta(t-t_n)/p}$$ and taking the supremum over $$t_n \leq t \leq \tau$$, we obtain   ‖X‖β,p,τ≤E[|X(tn)|p]1/p+cp1/pτ−tnE[|Y|p]1/p+cp1/pKp2β‖X‖β,p,τ. Choosing $$\beta$$ sufficiently large, i.e., $$c_p^{1/p}K\sqrt{p/2\beta} < 1$$, we secure the bound   ‖X‖β,p,τ≤11−cp1/pKp/2β(cp1/pτ−tnE[|Y|p]1/p+E[|X(tn)|p]1/p). The result follows upon multiplication by $$e^{\beta(\tau-t_n)/p}$$, since   eβ(τ−tn)/p‖X‖β,p,τ=(suptn≤t≤τeβ(τ−t)E[|X(t)|p])1/p≥suptn≤t≤τE[|X(t)|p]1/p. □ Estimate 6.12 Let $${\mathscr{T}}_5$$ be defined in Lemma 6.3. Then   T5=O(ε). Proof. This follows as in Karlsen & Storrøsten (2017, Limit 6). □ Estimate 6.13 Let $${\mathscr{T}}_6$$ be defined in Lemma 6.3. Then   |T6|≤2∑n=0N−1E[‖SCL(Δt)un−wn‖1,ϕ⋆Jr]. Proof. First, we note that $$\left|{S_\delta(b)-S_\delta(a)}\right| \le \left|{b-a}\right|$$. This and (4.15) yields   |T6|≤∑n=0N−1E[‖η~(tn+1)−η~((tn+1)−)‖1,ϕ⋆Jr]. Since   η~(tn+1)−η~((tn+1)−)=SSDE(tn+1,tn)(SCL(Δt)un−wn)+SCL(Δt)un−wn, the result follows from (3.2). □ Proof of Theorem 6.1. Consider Lemma 6.3 and take the upper limits in (6.6) as $$r_0 \downarrow 0, \varepsilon \downarrow 0$$ and $${\it {\gamma}} \downarrow 0$$ (in that order). Next we recall that $$w^n = w^n_k$$. Letting $$k \rightarrow \infty$$, then $$w^n_k \rightarrow {\mathcal{S}_{\text{CL}}}({{{\it {\Delta}} t}})u^n$$ in $$L^1({\it {\Omega}},L^1({\mathbb{R}}^d,\phi))$$. Because of the $$L^1$$-Lipschitz continuity of $${\mathcal{S}_{\text{CL}}}$$ (cf. Proposition 4.8) and the uniform BV-bound on the splitting approximation, it follows from Estimates 6.4–6.10 and 6.12–6.13 that   E[‖u0−u0‖1,ϕ]+Cϕ‖f‖Lip∫0t0E[‖ηΔt(t)−u(t)‖1,ϕ]dt +O(δ+r+Δt+δr+Δtδ)≥E[‖ηΔt(t0)−u(t0)‖1,ϕ]. Finally, we apply Grönwall’s inequality and then choose $$\delta = {{{\it {\Delta}} t}}^{2/3}$$ and $$r = {{{\it {\Delta}} t}}^{1/3}$$. □ A. Appendix A.1 Proof of Proposition 6.2 The proof of Proposition 6.2 is based on the following result. Lemma A.1 Suppose $$u,w \in L^2({\it {\Omega}},P,{\mathscr{F}}_{t_n};L^2({\mathbb{R}}^d))$$ and $$w$$ is smooth. Set   ψ(t)=(SSDE(t,tn)−I)w,v(t)=SCL(t−tn)u,t∈[tn,tn+1]. Then for all $$(S,Q) \in \mathscr{E}$$, all non-negative $${\varphi} \in C^\infty_c({\it {\Pi}}_n^2)$$ and all $$V \in \mathcal{S}$$,   R−L+T1+T2−T3+T4≥0, where   L =E[∬Πn∫RdS(v(tn+1,x)+ψ(s,y)−V)φ(tn+1,x,s,y)dxdyds] +E[∬Πn∫RdS(v(t,x)+ψ(tn+1,y)−V)φ(t,x,tn+1,y)dydxdt],R =E[∬Πn∫RdS(v(tn,x)+ψ(s,y)−V)φ(tn,x,s,y)dxdyds] +E[∬Πn∫RdS(v(t,x)+ψ(tn,y)−V)φ(t,x,tn,y)dydxdt],T1 =E[⨌Πn2S(v(t,x)+ψ(s,y)−V)(∂t+∂s)φdX],T2 =E[⨌Πn2Q(v(t,x),V−ψ(s,y))⋅∇xφdX],T3 =E[⨌Πn2S″(v(t,x)+ψ(s,y)−V)DsVσ(ψ(s,y)+w(y))φdX],T4 =12E[⨌Πn2S″(v(t,x)+ψ(s,y)−V)σ2(ψ(s,y)+w(y))φdX] and $${\it {\Pi}}_n = [t_n,t_{n+1}] \times {\mathbb{R}}^d$$. Proof. The entropy inequality reads    ∫RdS(v(tn,x)−c)φ(tn,x,s,y)−S(v(tn+1,x)−c)φ(tn+1,x,s,y)dx +∬ΠnS(v−c)∂tφ+Q(v,c)⋅∇xφdtdx≥0 (A.1) for all $$c \in {\mathbb{R}}$$ and all $$s,y \in {\it {\Pi}}_n$$. Specify $$c = V-\psi(s,y)$$ in (A.1), integrate in $$(s,y)$$ and take expectations to obtain   E[∬Πn∫RdS(v(tn,x)+ψ(s,y)−V)φ(tn,x,s,y)dxdsdy] −E[∬Πn∫RdS(v(tn+1,x)+ψ(s,y)−V)φ(tn+1,x,s,y)dxdsdy] +E[⨌Πn2S(v+ψ−V)∂tφ+Q(v,V−ψ)⋅∇xφdX]≥0. (A.2) Note that $$v(t)$$ is $${\mathscr{F}}_{t_n}$$-adapted for all $$t \in [t_n,t_{n+1}]$$. To reveal the equation satisfied by $$\psi$$, let $$\zeta(t) = {\mathcal{S}_{\text{SDE}}}(t,t_n)w$$. By definition,   ζ(t,x)=w(x)+∫tntσ(ζ(r,x))dB(r). Since $$\psi(t) = \zeta(t) - w$$,   ψ(t,x)=∫tntσ(ψ(r,x)+w(x))dB(r),t∈[tn,tn+1]. (A.3) Fix $$t,x \in {\it {\Pi}}_n, y \in {\mathbb{R}}^d$$ and set   X(s):=v(t,x)+ψ(s,y),F(X(s),V,s):=S(X(s)−V)φ(t,x,s,y),s∈[tn,tn+1]. By (A.3),   X(s)=v(t,x)+∫tnsσ(ψ(r,y)+w(y))dB(r). By Theorem A.10,   S(X(tn+1)−V)φ(t,x,tn+1,y) =S(X(tn)−V)φ(t,x,tn,y) +∫tntn+1S(X(s)−V)∂sφds +∫tntn+1S′(X(s)−V)σ(ψ(s)+w)φdB(s) −∫tntn+1S″(X(s)−V)DsVσ(ψ(s)+w)φds +12∫tntn+1S″(X(s)−V)σ2(ψ(s)+w)φds, where the stochastic integral is interpreted as a Skorohod integral. Upon integrating in $$t,x,y$$ and taking expectations,   E[∬Πn∫RdS(v(t,x)+ψ(tn,y)−V)φ(t,x,tn,y)dydtdx] −E[∬Πn∫RdS(v(t,x)+ψ(tn+1,y)−V)φ(t,x,tn+1,y)dydtdx] +E[⨌Πn2S(v(t,x)+ψ(s,y)−V)∂sφdX] +12E[⨌Πn2S″(v(t,x)+ψ(s,y)−V)(σ(ψ(s,y)+w(y)))2φdX] −E[⨌Πn2S″(v(t,x)+ψ(s,y)−V)DsVσ(ψ(s,y)+w(y))φdX]=0. (A.4) Adding (A.2) and (A.4) concludes the proof. □ Proof of Proposition 6.2. We use   φ(t,x,s,y)=12dϕ(t+s2,x+y2)Jr(x−y2)Jr0(t−s) (A.5) in Lemma A.1 and then send $$r_0, r$$ to zero (in that order). The sought-after result for $$V \in \mathcal{S}$$ is a consequence of Limits A.2–A.6 below. The extension to $$V \in {\mathbb{D}}^{1,2}$$ follows by an approximation argument as in Karlsen & StorrØsten (2017, Lemma 2.2). □ Limit A.2 Let $$L, R$$ be defined in Lemma A.1 and $${\varphi}$$ in (A.5). Then   limr,r0↓0L(r,r0) =E[∫RdS(v(tn+1,x)+ψ(tn+1,x))ϕ(tn+1,x)dx],limr,r0↓0R(r,r0) =E[∫RdS(v(tn,x)+ψ(tn,x)−V)ϕ(tn,x)dx]. Proof. Let us only consider the term   E[∬Πn∫RdS(v(tn+1,x)+ψ(s,y)−V)φ(tn+1,x,s,y)dxdyds]=:Z. The remaining terms can be treated in the same way. As a consequence of the dominated convergence theorem and Lemma A.9,    limr0↓0Z=12E[∫Rd∫RdS(v(tn+1,x)+ψ(tn+1,y)−V) ×12dϕ(tn+1,x+y2)Jr(x−y2)dxdy]. Moreover,   limr,r0↓0Z=12E[∫RdS(v(tn+1,x)+ψ(tn+1,x)−V)ϕ(tn+1,x)]. □ Limit A.3 Let $${\mathscr{T}}_1$$ be defined in Lemma A.1 and $${\varphi}$$ in (A.5). Then   limr,r0↓0T1=E[∬ΠnS(u(t,x)−V)∂tϕ(t,x)dxdt]. Proof. Observe that   (∂t+∂s)φ(t,x,s,y)=12d∂1ϕ(t+s2,x+y2)Jr(x−y2)Jr0(t−s). The result follows by the dominated convergence theorem and Lemma A.9; consult the proof of Limit A.2. □ Limit A.4 Let $${\mathscr{T}}_2$$ be defined in Lemma A.1 and $${\varphi}$$ in (A.5). Then   limr,r0↓0T2 =E[∬ΠnQ(v+ψ,V)⋅∇ϕdxdt] +E[∬Πn(∫Vv+ψS′(z−V)(f′(z−ψ)−f′(z))dz)⋅∇ϕdxdt] +E[∬Πn(∫Vv+ψS″(z−V)f′(z−ψ)dz)⋅∇ψϕdxdt]. Proof. First observe that   (∇x+∇y)φ(t,x,s,y)=12d∂2ϕ(t+s2,x+y2)Jr(x−y2)Jr0(t−s). Integration by parts results in   T2 =E[⨌Πn2Q(v(t,x),V−ψ(s,y)) ⋅12d∂2ϕ(t+s2,x+y2)Jr(x−y2)Jr0(t−s)dxdtdyds] +E[⨌Πn2∇y⋅Q(v(t,x),V−ψ(s,y))φ(t,x,s,y)dxdtdyds] =:T21+T22. It is straightforward to show that   limr,r0↓0T21 =E[∬ΠnQ(v(t,x),V−ψ(t,x))⋅∇ϕ(t,x)dxdt]. Finally, we apply the identity   Q(v,V−ψ)=Q(v+ψ,V)+∫Vv+ψS′(z−V)(f′(z−ψ)−f′(z))dz. Consider $${\mathscr{T}}_2^2$$. By the chain rule,   T22=−E[⨌Πn2∂2Q(v(t,x),V−ψ(s,y))⋅∇yψ(s,y)φ(t,x,s,y)dxdtdyds]. Sending $$r_0, r$$ to zero, we arrive at   limr,r0↓0T22=−E[∬Πn∂2Q(v(t,x),V−ψ(t,x))⋅∇xψ(t,x)ϕ(t,x)dxdt]. Finally, note that   ∂2Q(v,V−ψ) =−∫V−ψvS″(z−V+ψ)f′(z)dz =−∫Vv+ψS″(z−V)f′(z−ψ)dz. This concludes the proof. □ Limit A.5 Let $${\mathscr{T}}_3$$ be defined in Lemma A.1 and $${\varphi}$$ in (A.5). Then   limr,r0↓0T3=E[∬ΠnS″(v(t,x)+ψ(t,x)−V)DtVσ(ψ(t,x)+w(x))ϕ(t,x)dxdt]. Proof. The proof is a straightforward application of the dominated convergence theorem and Lemma A.9. □ Limit A.6 Let $${\mathscr{T}}_4$$ be defined in Lemma A.1 and $${\varphi}$$ by (A.5). Then   limr,r0↓0T4=12E[∬ΠnS″(v(t,x)+ψ(t,x)−V)σ2(ψ(t,x)+w(x))ϕ(t,x)dxdt]. Proof. This term may be treated similarly to $${\mathscr{T}}_3$$. □ A.2 Weighted $$L^p$$-spaces In the next two lemmas, we collect a few elementary properties of (weight) functions in $$\mathfrak{N}$$. For proofs, see Karlsen & Storrøsten (2017). Lemma A.7 Suppose $$\phi \in \mathfrak{N}$$ and $$0 < p < \infty$$. Then, for $$x,z\in {\mathbb{R}}^d$$,   |ϕ1/p(x+z)−ϕ1/p(x)|≤wp,ϕ(|z|)ϕ1/p(x), where   wp,ϕ(r)=Cϕpr(1+CϕpreCϕr/p), which is defined for all $$r \geq 0$$. As a consequence it follows that if $$\phi(x_0) = 0$$ for some $$x_0 \in {\mathbb{R}}^d$$ then $$\phi \equiv 0$$ (and by definition $$\phi \notin \mathfrak{N}$$). Lemma A.8 Fix $$\phi \in \mathfrak{N}$$, and let $$w_{p,\phi}$$ be defined in Lemma A.7. Let $$J$$ be a mollifier as defined in Section 2 and take $$\phi_\delta = \phi \star J_\delta$$ for $$\delta > 0$$. Then (i) $$\phi_\delta \in \mathfrak{N}$$ with $$C_{\phi_\delta} = C_\phi$$; (ii) for any $$u \in L^p({\mathbb{R}}^d,\phi)$$,   |‖u‖p,ϕp−‖u‖p,ϕδp|≤w1,ϕ(δ)min{‖u‖p,ϕp,‖u‖p,ϕδp}; (ii)   |Δϕδ(x)|≤1δCϕ‖∇J‖L1(Rd)(1+w1,ϕ(δ))2ϕδ(x). A.3 A ‘doubling-of-variables’ tool The following result follows along the lines of Serre (1999, Lemma 2.7.2). See also Karlsen & Storrøsten (2017, Section 6). Lemma A.9 Suppose $$u,v \in L^1_{\mathrm{loc}}({\mathbb{R}}^d)$$ and $$F$$ is Lipschitz on $${\mathbb{R}}^2$$. Fix $$\psi \in C_c({\mathbb{R}}^d)$$ and set   Tr :=∫Rd∫RdF(u(x),v(y))12dψ(x+y2)Jr(x−y2)dydx −∫RdF(u(x),v(x))ψ(x)dx, where $$J_r$$ is defined in (2.2). Then $$\mathcal{T}_r \rightarrow 0$$ as $$r \downarrow 0$$. Similarly, let $$G:[0,T] \times {\mathbb{R}} \rightarrow {\mathbb{R}}$$ be measurable in the first variable and Lipschitz continuous in the second variable. With $$w \in L^1([0,T])$$, set   Tr0(s)=∫0T|G(s,w(t))−G(s,w(s))|Jr0(t−s)dt. Then $$\mathcal{T}_{r_0}(s) \rightarrow 0$$ for a.e. $$s$$ as $$r_0 \downarrow 0$$. The above results do not rely on the symmetry of $$J$$. A.4 A version of Itô ’s formula Here we recall the particular anticipating Itô formula applied in the proof of Lemma 5.1 and Lemma A.1. The proof of this follows Nualart (2006, Theorem 3.2.2) closely. However, because of the particular assumptions, certain points simplify. See Karlsen & Storrøsten (2017, Theorem 6.7) for an outline of a proof. Theorem A.10 Let $$X$$ be a continuous process of the form   X(t)=X0+∫0tu(s)dB(s)+∫0tv(s)ds, where $$u:[0,T] \times {\it {\Omega}} \rightarrow {\mathbb{R}}$$ and $$v:[0,T] \times {\it {\Omega}} \rightarrow {\mathbb{R}}$$ are predictable processes, satisfying   E[(∫0Tu2(s,z)ds)2<∞,E[∫0Tv2(s)ds]]<∞, and $$X_0 \in L^2({\it {\Omega}},{\mathscr{F}}_0,P)$$. Let $$F:{\mathbb{R}}^2 \times [0,T] \rightarrow {\mathbb{R}}$$ be twice continuously differentiable. Suppose there exists a constant $$C > 0$$ such that for all $$(\zeta,\lambda,t) \in {\mathbb{R}}^2 \times [0,T]$$,   |F(ζ,λ,t)|,|∂3F(ζ,λ,t)|≤C(1+|ζ|+|λ|),|∂1F(ζ,λ,t)|,|∂1,22F(ζ,λ,t)|,|∂12F(ζ,λ,t)|≤C. Let $$V \in {\mathcal{S}}$$. Then $$s \mapsto \partial_1F(X(s),V,s)u(s)$$ is Skorohod integrable and   F(X(t),V,t) =F(X0,V,0) +∫0t∂3F(X(s),V,s)ds +∫0t∂1F(X(s),V,s)u(s,z)dB(s) +∫0t∂1F(X(s),V,s)v(s)ds +∫0t∂1,22F(X(s),V,s)DsVu(s)ds +12∫0t∂12F(X(s),V,s)u2(s)ds,dP-almost surely, where the stochastic integral is interpreted as a Skorohod integral. A.5 Young measures The purpose of this section is to provide a reference for some results concerning Young measures and their use in representation formulas for weak limits. For a more general introduction (see, e.g., (see, e.g., Valadier, 1994; Málek et al., 1996; Florescu & Godet-Thobie, 2012). Let $$(X,\mathscr{A},\mu)$$ be a $$\sigma$$-finite measure space and $$\mathscr{P}({\mathbb{R}})$$ the set of probability measures on $${\mathbb{R}}$$. In this article, $$X$$ is typically $${\it {\Pi}}_T \times {\it {\Omega}}$$. A Young measure from $$X$$ into $${\mathbb{R}}$$ is a function $$\nu:X \rightarrow \mathscr{P}({\mathbb{R}})$$ such that $$x \mapsto \nu_x(B)$$ is $$\mathscr{A}$$-measurable for every Borel measurable set $$B \subset {\mathbb{R}}$$. We denote by $$\mathcal{Y}\left({X,\mathscr{A},\mu;{\mathbb{R}}}\right)$$, or $$\mathcal{Y}\left({X;{\mathbb{R}}}\right)$$ if the measure space is understood, the set of all Young measures from $$X$$ into $${\mathbb{R}}$$. The following theorem is proved in Pedregal (1997, Theorem 6.2) in the case that $$X \subset {\mathbb{R}}^n$$ and $$\mu$$ is the Lebesgue measure. Theorem A.11 Fix a $$\sigma$$-finite measure space $$(X,\mathscr{A},\mu)$$. Let $$\zeta:[0,\infty) \rightarrow [0,\infty]$$ be a continuous, nondecreasing function satisfying $$\lim_{\xi \rightarrow \infty}\zeta(\xi) = \infty$$ and $$\left\{{u^n}\right\}_{n \ge 1}$$ a sequence of measurable functions such that   supn∫Xζ(|un|)dμ(x)<∞. Then there exist a subsequence $$\left\{{u^{n_j}}\right\}_{j \ge 1}$$ and $$\nu \in \mathcal{Y}\left({X,\mathscr{A},\mu;{\mathbb{R}}}\right)$$ such that for any Carathéodory function $$\psi:{\mathbb{R}} \times X \rightarrow {\mathbb{R}}$$ with $$\psi(u^{n_j}(\cdot),\cdot) \rightharpoonup \overline{\psi}$$ in $$L^1(X)$$, we have   ψ¯(x)=∫Rψ(ξ,x)dνx(ξ). The proof is based on the embedding of $$\mathcal{Y}\left({X;{\mathbb{R}}}\right)$$ into $$L^\infty_{w*}(X,{\mathbb{R}}ad{{\mathbb{R}}})$$. Here $${\mathbb{R}}ad{{\mathbb{R}}}$$ denotes the space of Radon measures on $${\mathbb{R}}$$. The crucial observation is that $$(L^1(X,C_0({\mathbb{R}})))^*$$ is isometrically isomorphic to $$L^\infty_{w*}(X,{\mathbb{R}}ad{{\mathbb{R}}})$$ also in the case that $$(X,\mathscr{A},\mu)$$ is an abstract $$\sigma$$-finite measure space. It is relatively straightforward to go through the proof and extend it to the more general case (Málek et al., 1996, Theorem 2.11). The result then follows as an application of Alaoglu’s theorem combined with the Eberlein-Šmulian theorem. Note, however, because of our use of weighted $$L^p$$-spaces, it suffices to use the version for finite measure spaces. A.6 Weak compactness in $$L^1$$. To apply Theorem A.11 it is necessary to know whether $$\left\{{\psi(\cdot,u^n(\cdot))}\right\}_{n \ge 1}$$ has a subsequence converging weakly in $$L^1(X)$$. The key result is the well-known Dunford–Pettis theorem. Definition A.12 Let $$\mathcal{K} \subset L^1(X,\mathscr{A},\mu)$$. (i) $$\mathcal{K}$$ is uniformly integrable if for any $$\varepsilon > 0$$ there exists $$c_0(\varepsilon)$$ such that   supf∈K∫|f|≥c|f|dμ≤ε whenever c≥c0(ε). (ii) $$\mathcal{K}$$ has a uniform tail if for any $$\varepsilon > 0$$ there exists $$E \in \mathscr{A}$$ with $$\mu(E) < \infty$$ such that   supf∈K∫X∖E|f|dμ≤ε. If $$\mathcal{K}$$ satisfies both (i) and (ii) it is said to be equiintegrable. Remark A.13 Note that (ii) is void when $$\mu$$ is finite. Theorem A.14 (Dunford–Pettis) Let $$(X,\mathscr{A},\mu)$$ be a $$\sigma$$-finite measure space. A subset $$\mathcal{K}$$ of $$L^1(X)$$ is relatively weakly sequentially compact if and only if it is equiintegrable. There are a couple of well-known reformulations of uniform integrability. Lemma A.15 Suppose $$\mathcal{K} \subset L^1(X)$$ is bounded. Then $$\mathcal{K}$$ is uniformly integrable if and only if (i) for any $$\varepsilon > 0$$ there exists $$\delta(\varepsilon) > 0$$ such that   supf∈K∫E|f|dμ≤ε whenever μ(E)≤δ(ε); (ii) there is an increasing function $$\Psi:[0,\infty) \rightarrow [0,\infty)$$ such that $$\Psi(\zeta)/\zeta \rightarrow \infty$$ as $$\zeta \rightarrow \infty$$ and   supf∈K∫XΨ(|f(x)|)dμ(x)<∞. Acknowledgements We are grateful to N. H. Risebro and an anonymous referee for many valuable comments. Funding Research Council of Norway through the project Stochastic Conservation Laws (250674/F20). References Bauzet C. ( 2015) Time-splitting approximation of the Cauchy problem for a stochastic conservation law. Math. Comput. Simulat. , 118, 73– 86. Google Scholar CrossRef Search ADS   Bauzet C., Charrier J. & Gallouët T. ( 2016) Convergence of flux-splitting finite volume schemes for hyperbolic scalar conservation laws with a multiplicative stochastic perturbation. Math. Comp. , 85, 2777– 2813. 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Modern Birkhäuser Classics. Birkhäuser/Springer, New York. Google Scholar CrossRef Search ADS   Debussche A., Hofmanová M. & Vovelle J. ( 2016) Degenerate parabolic stochastic partial differential equations: Quasilinear case. Ann. Probab. , 44, 1916– 1955. Google Scholar CrossRef Search ADS   Debussche A. & Vovelle J. ( 2010) Scalar conservation laws with stochastic forcing. J. Funct. Anal. , 259, 1014– 1042. Google Scholar CrossRef Search ADS   Debussche A. & Vovelle J. ( 2015) Invariant measure of scalar first-order conservation laws with stochastic forcing. Probab. Theory Related Fields , 163, 575– 611. Google Scholar CrossRef Search ADS   Eymard R., Gallouët T. & Herbin R. ( 2000) Finite volume methods. Handbook of Numerical Analysis  ( Ciarlet P. G. & Lions J. L. eds), vol. VII. Amsterdam: North-Holland, pp. 713– 1020. Feng J. & Nualart D. ( 2008) Stochastic scalar conservation laws. J. Funct. Anal. , 255, 313– 373. Google Scholar CrossRef Search ADS   Florescu L. C. & Godet-Thobie C. ( 2012) Young Measures and Compactness in Measure Spaces . Berlin: De Gruyter. Google Scholar CrossRef Search ADS   Gess B. & Souganidis P. E. ( 2016) Long-time behavior, invariant measures and regularizing effects for stochastic scalar conservation laws. Comm. Pure. and Appl. Math. , doi: 10.1002/cpa.21646. Gess B. & Souganidis P. E. ( 2015) Scalar conservation laws with multiple rough fluxes. Commun. Math. Sci. , 13, 1569– 1597. Google Scholar CrossRef Search ADS   Hofmanová M. ( 2013) Degenerate parabolic stochastic partial differential equations. Stochastic Process. Appl. , 123, 4294– 4336. Google Scholar CrossRef Search ADS   Holden H., Karlsen K. H., Lie K.-A. & Risebro N. H. ( 2010) Splitting Methods for Partial Differential Equations with Rough Solutions: Analysis and MATLAB programs . EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society (EMS). Google Scholar CrossRef Search ADS   Holden H. & Risebro N. H. ( 1997) Conservation laws with random source. Appl. Math. Optim . 36, 229– 241. Google Scholar CrossRef Search ADS   http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Numerical Analysis Oxford University Press

Analysis of a splitting method for stochastic balance laws

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Abstract

Abstract We analyse a semidiscrete splitting method for conservation laws driven by a semilinear noise term. Making use of fractional bounded variation (BV) estimates, we show that the splitting method generates approximate solutions converging to the exact solution, as the time step $$\Delta t \to 0$$. Under the assumption of a homogenous noise function, and thus the availability of BV estimates, we provide an $$L^1$$-error estimate. Bringing into play a generalization of Kružkov’s entropy condition, permitting the ‘Kružkov constants’ to be Malliavin differentiable random variables, we establish an $$L^1$$-convergence rate of order $$\frac13$$ in $$\Delta t$$. 1. Introduction Recently, there have been many works studying the effect of stochastic forcing on scalar conservation laws (Vallet, 2000; Kim, 2003; Feng & Nualart, 2008; Vallet & Wittbold, 2009; Debussche & Vovelle, 2010, 2015; Bauzet et al., 2012; Chen et al., 2012; Hofmanov´a, 2013; Biswas et al., 2015; Karlsen & Storrøsten, 2017; Debussche & Vovelle, 2015; Debussche et al., 2016) with emphasis on existence, uniqueness and stability questions. Deterministic conservation laws exhibit shocks (discontinuous solutions), and a weak formulation coupled with an appropriate entropy condition is required to ensure the well-posedness (Kružkov, 1970). The question of uniqueness gets somewhat more difficult by adding a stochastic source term due to the interaction between noise and nonlinearity. A pathwise theory for conservation laws with stochastic fluxes has been developed in Lions et al. (2013, 2014) and Gess & Souganidis (2014, 2015). In this article, we are interested in the convergence of approximate solutions to conservation laws driven by a multiplicative Wiener noise term, i.e., stochastic balance laws of the form   du+divf(u)dt=σ(x,u)dB,(t,x)∈ΠT (1.1) with initial data   u(0,x)=u0(x),x∈Rd. (1.2) We denote by $$\nabla$$ and $${\operatorname{div}}=\nabla\cdot$$ the spatial gradient and divergence, respectively. Moreover, $${\it {\Pi}}_T = {\mathbb{R}}^d \times (0,T)$$ for some fixed final time $$T>0$$, and $$u(x,t)$$ is the scalar unknown function that is sought. The random force in (1.1) is driven by a Wiener process $$B=B(t)=B(t,\omega)$$, $$\omega\in {\it {\Omega}}$$, over a stochastic basis $$({\it {\Omega}},{\mathscr{F}},\left\{{{\mathscr{F}}_t}\right\}_{t\ge 0},P)$$, where $$P$$ is a probability measure, $${\mathscr{F}}$$ is a $$\sigma$$-algebra and $$\left\{{{\mathscr{F}}_t}\right\}_{t\ge 0}$$ is a right-continuous filtration on $$({\it {\Omega}},{\mathscr{F}})$$ such that $${\mathscr{F}}_0$$ contains all the $$P$$-negligible subsets. The convection flux $$f:{\mathbb{R}}\to{\mathbb{R}}^d$$ satisfies   f is (globally) Lipschitz continuous on R. (𝓐f) Furthermore, we will sometimes make use of the assumption   f″ is uniformly bounded on R. (𝓐f,1) The noise coefficient $$\sigma:{\mathbb{R}}^d\times {\mathbb{R}}\to {\mathbb{R}}$$ is assumed to satisfy   ‖σ‖Lip=supx∈Rdsupu≠v{|σ(x,u)−σ(x,v)||u−v|}<∞,|σ(⋅,0)|∈L∞(Rd). (𝓐σ) These assumptions imply   |σ(x,u)−σ(x,v)|≤‖σ‖Lip|u−v|,|σ(x,u)|≤max{‖σ‖Lip,‖σ(⋅,0)‖L∞(Rd)}(1+|u|). Furthermore, we often assume the existence of constants $$M_\sigma$$ and $${\kappa_\sigma}$$ such that   |σ(x,u)−σ(y,u)|≤Mσ|x−y|κσ+1/2(1+|u|),κσ∈(0,1/2]. (𝓐σ,1) A prevailing difficulty affecting convergence/error analysis is related to the time discretization and the interplay between noise and nonlinearity. Up to now, there have been only a few studies investigating this problem. Holden & Risebro (1997) study a one-dimensional equation with bounded initial data and a compactly supported, homogeneous noise function $$\sigma=\sigma(u)$$, ensuring $$L^\infty$$-bounds on the solution. An operator splitting method is used to construct approximate solutions, and it is shown that a subsequence of these approximations converges to a (possibly nonunique) weak solution. Recently, this work was generalized to stochastic entropy solutions and extended to the multidimensional case by Bauzet (2015). Kröker & Rohde (2012) analyse semidiscrete (time-continuous) finite volume methods. They use the compensated compactness method to prove convergence to a stochastic entropy solution for one-dimensional equations, with nonhomogeneous noise function $$\sigma=\sigma(x,u)$$. Bauzet et al. (2016) analyse fully discrete finite volume methods for multidimensional equations, with homogeneous noise function $$\sigma=\sigma(u)$$. Their proof relies on weak bounded variation (BV) (energy) estimates and a uniqueness result for measure-valued stochastic entropy solutions. In this article (as in Holden & Risebro, 1997; Bauzet, 2015), we will investigate the semidiscrete splitting method for calculating approximations to stochastic entropy solutions of (1.1). Generally, this method is based on ‘splitting off’ the effect of the stochastic source $$\sigma(x,u)\, \mathrm{d} B$$. This Godunov-type operator splitting can be used to extend sophisticated numerical methods for deterministic conservation laws to stochastic balance laws. Generally, the tag ‘operator splitting’ refers to the well-known idea of constructing numerical methods for complicated partial differential equations by reducing them to a progression of simpler equations, each of which can be solved by some tailor-made numerical method. The operator splitting approach is described in a large number of articles and books. We do not survey the literature here, referring the reader instead to the bibliography in Holden et al. (2010). The main focus of the book (Holden et al., 2010) is on convergence results, within classes of discontinuous functions, for general splitting algorithms for deterministic nonlinear partial differential equations. Compared with the earlier results of Holden & Risebro (1997) and Bauzet (2015), the main contributions of this article are twofold. First, we establish convergence of the splitting approximations to a stochastic entropy solution in the case of nonhomogeneous noise functions $$\sigma = \sigma(x,u)$$. Whenever $$\sigma$$ has a dependency on the spatial position $$x$$, BV estimates are no longer available and the approach resorted to in Holden & Risebro (1997) and Bauzet (2015) does not apply. Following an idea laid out in Chen et al. (2012), and independently in Debussche & Vovelle (2010), we derive a fractional $$BV_x$$ estimate, which, via an interpolation argument à la Kružkov, is turned into a temporal equicontinuity estimate. These a priori estimates, along with Young measures and an earlier uniqueness result, are used to show that splitting approximations converge to a stochastic entropy solution. Let us make a few comments about the convergence proof. In the deterministic case, the spatial and temporal estimates would imply strong ($$L^1$$) compactness of the splitting approximations. In the stochastic setting, we have the randomness variable $$\omega$$ for which there is no compactness; as a matter of fact, possible ‘oscillations’ in $$\omega$$ may prevent strong compactness. In the literature, the standard way of dealing with this issue is to look for tightness (weak compactness) of the probability laws of the approximations. Then an application of the Skorokhod representation theorem provides a new probability space and new random variables, with the same laws as the original variables, that do converge strongly (almost surely) in $$\omega$$ to some limit. Equipped with almost sure convergence, it is not difficult to show that the limit variable is a so-called martingale solution, i.e., the limit is probabilistic weak in the sense that the stochastic basis is now viewed as part of the solution. One can pass (à la Yamada & Watanabe) from martingale to pathwise solutions provided there is a strong uniqueness result. In this article, we will not follow this ‘traditional’ approach. Instead, we will utilize Young measures, parametrized over $$(t,x,\omega)$$, to represent weak limits of nonlinear functions, thereby obtaining weak convergence of the splitting approximations toward a so-called Young measure-valued stochastic entropy solution. We use the spatial and temporal translation estimates to conclude that the limit is a solution in this sense. Weak convergence is then upgraded to strong convergence in $$(t,x,\omega)$$a posteriori, thanks to the fact that these measure-valued solutions are $$L^1$$-stable (unique). After the works of Tartar, DiPerna and others, weak compactness arguments of this type (propagation of compactness) are frequently used in the nonlinear partial differential equations (PDE) literature (cf., e.g., Szepessy, 1989; Málek et al., 1996; Panov, 1996; Eymard et al., 2000) and recently in the context of stochastic equations (Vallet & Wittbold, 2009; Bauzet et al., 2012, 2016; Bauzet, 2015; Biswas et al., 2015; Karlsen & Storrøsten, 2017). Our second main contribution is an $$L^1$$-error estimate of the form $${\mathcal O}(\Delta t^{1/3})$$, for homogeneous noise functions $$\sigma=\sigma(u)$$. Except for the expected convergence rate for the vanishing viscosity method (Chen et al., 2012), this appears to be the first error estimate derived for approximate solutions to stochastic conservation laws. The rate $$\frac13$$ should be compared with the first-order convergence rate available for conservation laws with deterministic source (Langseth et al., 1996). Our proof relies on BV estimates and a generalization of the Kružkov entropy condition, allowing the ‘Kružkov constants’ to be Malliavin differentiable random variables, which was put forward in the recent work Karlsen & Storrøsten (2017). The remaining part of this article is organized as follows: Section 2 collects some preliminarily material along with the relevant notion of (stochastic entropy) solution. The operator splitting method is defined precisely in Section 3. A series of a priori estimates are derived in Section 4, which are subsequently used in Section 5 to prove convergence toward a stochastic entropy solution. Section 6 is devoted to the proof of the error estimate. Section A is an appendix collecting some definitions and results used elsewhere in the paper. 2. Preliminaries In this article, as in Karlsen & Storrøsten (2017), we apply certain weighted $$L^p$$ spaces. Since we do not assume $$\sigma(x,0) \equiv 0$$, weighted spaces on $${\mathbb{R}}^d$$ provide a convenient alternative to working on the torus as in Debussche & Vovelle (2010) and Debussche et al. (2016). The weights used herein turn out to be suitable also for fractional $$BV_x$$ estimates; cf. Proposition 4.4. Let $$\mathfrak{N}$$ be the set of all nonzero $$\phi \in C^1({\mathbb{R}}^d) \cap L^1({\mathbb{R}}^d)$$ for which there exists a constant $$C$$ such that $$\left|{\nabla\phi}\right| \le C \phi$$. An example is $$\phi(x) = \exp^{-\sqrt{1 + \left|{x}\right|^2}}$$. Set   Cϕ=inf{C∣||∇ϕ|≤Cϕ}. For $$\phi \in \mathfrak{N}$$, we use the weighted $$L^p$$-norm $$\left\Vert {\cdot}\right\Vert_{p,\phi}$$ defined by   ‖u‖p,ϕ:=(∫Rd|u(x)|pϕ(x)dx)1/p. The corresponding weighted $$L^p$$-space is denoted by $$L^p({\mathbb{R}}^d,\phi)$$. Similarly, we define   XXX‖u‖∞,ϕ−1:=supx∈Rd{|u(x)|ϕ(x)},u∈C(Rd). (2.1) Some useful results regarding functions in $$\mathfrak{N}$$ are collected in Section A.2. We denote by $$\mathscr{E}$$ the set of non-negative convex functions in $$C^2({\mathbb{R}})$$ such that $$S'$$ is bounded and $$S''$$ compactly supported. A pair of functions $$(S,Q) $$ is called an entropy/entropy–flux pair if $$S:{\mathbb{R}}\to{\mathbb{R}}$$ is $$C^2$$ and $$Q=(Q_1,\ldots,Q_d):{\mathbb{R}} \mapsto{\mathbb{R}}^d $$ satisfies $$Q' = S' f'$$. An entropy/entropy–flux pair $$(S,Q)$$ is said to belong to $$\mathscr{E}$$ if $$S$$ belongs to $$\mathscr{E}$$. Let $${\mathscr{P}}$$ denote the predictable $$\sigma$$-algebra on $$[0,T] \times {\it {\Omega}}$$ with respect to $$\left\{{{\mathscr{F}}_t}\right\}$$ (see, e.g., (see, e.g., Chung & Williams, 2014, Section 2.2). In general we are working with equivalence classes of functions with respect to the measure $$\mathrm{d}t \otimes {\rm d}P$$. The equivalence class $$u$$ is said to be predictable if it has a version $$\tilde{u}$$ that is $${\mathscr{P}}$$-measurable. Equivalently, we could ask for any representative to be $${\mathscr{P}}^*$$-measurable, where $${\mathscr{P}}^*$$ is the completion of $${\mathscr{P}}$$ with respect to $$\mathrm{d}t \otimes {\rm d}P$$. Note that any (jointly) measurable and adapted process is $${\mathscr{P}}^*$$-measurable (cf., e.g., Chung & Williams, 2014, Theorem 3.7). Next we collect some basic material related to Malliavin calculus. We refer to Nualart (2006) for an introduction to the topic. The Malliavin calculus is developed with respect to the isonormal Gaussian process $$W:L^2([0,T]) \rightarrow \mathcal{H}^1$$, defined by $$W(h) := \int_0^Th\,\mathrm{d}B$$. Here $$\mathcal{H}^1$$ is the subspace of $$L^2({\it {\Omega}},{\mathscr{F}},P)$$ consisting of zero-mean Gaussian random variables. We denote by $$\mathcal{S}$$ the class of smooth random variables of the form   V=f(W(h1),…,W(hn)), where $$f \in C^\infty_c({\mathbb{R}}^n)$$, $$h_1, \dots, h_n \in L^2([0,T])$$ and $$n \geq 1$$. For such random variables, the Malliavin derivative is defined by   DV=∑i=1n∂if(W(h1),…,W(hn))hi, where $$\partial_i$$ denotes the derivative with respect to the $$i$$th variable. The space $$\mathcal{S}$$ is dense in $$L^2({\it {\Omega}},{\mathscr{F}},P)$$. Furthermore, the operator $$D$$ is closable as a map from $$L^2({\it {\Omega}})$$ to $$L^2({\it {\Omega}};L^2([0,T]))$$ (Nualart, 2006, Proposition 1.2.1). The domain of $$D$$ in $$L^2({\it {\Omega}})$$ is denoted by $${\mathbb{D}}^{1,2}$$. That is, $${\mathbb{D}}^{1,2}$$ is the closure of $$\mathcal{S}$$ with respect to the norm   ‖V‖D1,2={E[|V|2]+E[‖DV‖L2([0,T])2]}1/2. For the generalization of the above to Hilbert space-valued random variables (see Nualart, 2006, Remark 2, p. 31). We use the notion of stochastic entropy solution introduced in Karlsen & Storrøsten (2017), which is a refinement of the notion introduced by Feng & Nualart (2008). Definition 2.1 Fix $$\phi \in \mathfrak{N}$$. A stochastic entropy solution $$u$$ of (1.1) and (1.2) with $$u_0 \in L^2({\it {\Omega}},{\mathscr{F}}_0,P;L^2({\mathbb{R}}^d,\phi))$$ is a stochastic process   u={u(t)=u(t,x)=u(t,x;ω)}t∈[0,T] satisfying the following conditions: (i) $$u$$ is a predictable process in $$L^2([0,T] \times {\it {\Omega}};L^2({\mathbb{R}}^d,\phi))$$. (ii) For any random variable $$V \in {\mathbb{D}}^{1,2}$$ and any entropy, entropy–flux pair $$(S,Q) \in \mathscr{E}$$,   E[∬ΠTS(u−V)∂tφ+Q(u,V)⋅∇φdxdt+∫RdS(u0(x)−V)φ(0,x)dx] −E[∬ΠTS″(u−V)σ(x,u)DtVφdxdt] +12E[∬ΠTS″(u−V)σ(x,u)2φdxdt]≥0 for all non-negative $$\varphi \in C^\infty_c([0,T) \times {\mathbb{R}}^d)$$. Here, $$L^2([0,T] \times {\it {\Omega}};L^2({\mathbb{R}}^d,\phi))$$ denotes the Lebesgue–Bochner space and $$D_tV$$ denotes the Malliavin derivative of $$V$$ evaluated at time $$t$$. By Karlsen & Storrøsten (2017, Lemma 2.2), it suffices to consider $$V \in \mathcal{S}$$ in (ii). In Karlsen & Storrøsten (2017), the existence and uniqueness of entropy solutions in the sense of Definition 2.1 is established under assumptions (𝓐f), (𝓐σ) and (𝓐σ,1). We also mention that whenever $$u_0 \in L^p({\it {\Omega}};L^p({\mathbb{R}}^d,\phi))$$ with $$2 \le p < \infty$$,   ess sup0≤t≤T{E[‖u(t)‖p,ϕp]}<∞. Let $$\left\{{J_\delta}\right\}_{\delta > 0}$$ be a sequence of symmetric mollifiers on $${\mathbb{R}}^d$$, i.e.,   Jδ(x)=1δdJ(xδ), (2.2) where $$J \geq 0$$ is a smooth, symmetric function satisfying $$\text{supp}\, (J)\subset B(0,1)$$ and $$\int J=1$$. For $$d = 1$$, we set $$J^+(x) = J(x - 1)$$, so that $$\text{supp}\,(J^+) \subset (0,2)$$. Under the additional assumption (𝓐σ,1), Karlsen & Storrøsten (2017, Proposition 5.2) assert that the entropy solution $$u$$ satisfies   E[∬Rd×Rd|u(t,x+z)−u(t,x−z)|Jr(z)ϕ(x)dx] ≤eCϕ‖f‖LiptE[∬Rd×Rd|u0(x+z)−u0(x−z)|Jr(z)ϕ(x)dx]+O(rκσ), (2.3) where $${{\kappa_\sigma}}$$ is given in (𝓐σ,1). Whenever $$\sigma(x,u) = \sigma(u)$$, the last term on the right-hand side vanishes, i.e., $${\mathcal{O}}(\ldots) = 0$$. 3. Operator splitting We will now describe the basic operator splitting method for (1.1). Let $$\mathcal{S}_{\text{CL}}(t)$$ be the solution operator that maps an initial function $$v_0(x)$$ to the unique entropy solution of the deterministic conservation law   ∂tv+divf(v)=0,v(0,x)=v0(x), (3.1) i.e., if $$v(t) := \mathcal{S}_{\text{CL}}(t)v_0$$ then $$v$$ is the unique entropy solution of (3.1). More precisely, for each $$\tau \in [0,T]$$,    ∫Rd|v0(x)−c|φ(0,x)dx−∫Rd|v(τ)−c|φ(τ,x)dx +∫0τ∫Rd|v−c|∂tφ+sign(v−c)(f(v)−f(c))⋅∇φdxdt≥0 for all $$c \in {\mathbb{R}}$$ and all non-negative $$\varphi \in C^\infty_c([0,T) \times {\mathbb{R}})$$. Note that the integrals are well defined due to the global Lipschitz assumption (𝓐f). Recall that the entropy solution has a version that belongs to $$C([0,T];L^1_{\mathrm{loc}}({\mathbb{R}}^d))$$ (Cancès & Gallouët, 2011). As we frequently need to consider the evaluation $$v(t),$$ it is convenient for us to assume that $$v$$ has this property. Let $$u,v \in L^1({\mathbb{R}}^d,\phi),$$ where $$\phi \in \mathfrak{N}$$. Then, for any $$t \in [0,T]$$,   ‖SCL(t)v−SCL(t)u‖1,ϕ≤eCϕ‖f‖Lipt‖u−v‖1,ϕ. Suppose $$u \in L^1({\it {\Omega}},{\mathscr{F}}_s,P;L^1({\mathbb{R}}^d,\phi))$$ for some $$s \in [0,T]$$. Let $$s \leq t \leq T$$. By considering the composition $${\it {\Omega}} \ni \omega \mapsto u(\omega) \mapsto \mathcal{S}_{\text{CL}}(t-s)u(\omega)$$, it follows that $$\mathcal{S}_{\text{CL}}(t-s)u$$ is $${\mathscr{F}}_s$$-measurable as an element in $$L^1({\mathbb{R}}^d,\phi)$$ (cf. Mishra & Schwab, 2012, Section 3.3). Similarly, for $$s \leq t \leq T$$, we let $${\mathcal{S}_{\text{SDE}}}(t,s)$$ denote the two-paramater semigroup defined by $${\mathcal{S}_{\text{SDE}}}(t,s)w^s = w(t)$$, where $$w$$ is the strong solution of   w(t,x)=ws(x)+∫stσ(x,w(r,x))dB(r). Suppose $$w^s,v^s \in L^1({\it {\Omega}},{\mathscr{F}}_s,P;L^1({\mathbb{R}}^d,\phi))$$. Then,   E[‖SSDE(t,s)ws−SSDE(t,s)vs‖1,ϕ]=E[‖ws−vs1,ϕ‖]. (3.2) To see this, let $$S_\delta \rightarrow \left|{\cdot}\right|$$ as $$\delta \downarrow 0$$ and consider the quantity $$S_\delta(w(t,x)-v(t,x))$$. Next, apply Itô’s formula, multiply by $$\phi$$ and let $$\delta \downarrow 0$$. Because of (3.2),   SSDE(⋅,s):L1(Ω,Fs,P;L1(Rd,ϕ))→L1([s,T]×Ω,P[s,T],dt⊗dP;L1(Rd,ϕ)), where $${\mathscr{P}}_{[s,T]}$$ denotes the predictable $$\sigma$$-algebra relative to $$\left\{{{\mathscr{F}}_t}\right\}_{s \le t \le T}$$ on $$[s,T] \times {\it {\Omega}}$$. Fix $$N \in {\mathbb{N}}$$, specify $${{{\it {\Delta}} t}} = T/N$$ and set $$t_n = n{{{\it {\Delta}} t}}$$. Let $$u^0 = u^0(x;\omega)$$ be given. The operator splitting, with initial condition $$u^0$$, is the sequence $$\left\{{u^n = u^n(x;\omega)}\right\}_{n = 0}^N$$ defined recursively by   un+1(x;ω)=[SSDE(tn+1,tn;ω)∘SCL(Δt)]un(x;ω) (3.3) for $$n = 0,1, \dots ,N-1$$. A graphical representation is given in Fig. 1. Fig. 1. View largeDownload slide A graphical representation of $$\left\{{u^n}\right\}$$, $${u_{{{{\it {\Delta}} t}}}}$$ and $${v_{{{{\it {\Delta}} t}}}}$$. Fig. 1. View largeDownload slide A graphical representation of $$\left\{{u^n}\right\}$$, $${u_{{{{\it {\Delta}} t}}}}$$ and $${v_{{{{\it {\Delta}} t}}}}$$. To investigate the convergence of the semidiscrete splitting algorithm (3.3), we need to work with functions that are defined not only for each $$t_n = n{{{\it {\Delta}} t}}$$ but also for the entire interval $$[0,T]$$. To this end, we introduce two different ‘time interpolants’ $${u_{{{{\it {\Delta}} t}}}}(t)={u_{{{{\it {\Delta}} t}}}}(t,x;\omega)$$ and $${v_{{{{\it {\Delta}} t}}}}(t)={v_{{{{\it {\Delta}} t}}}}(t,x;\omega)$$, defined for $$n=0,\ldots,N-1$$ by   uΔt(t)=SSDE(t,tn)∘SCL(Δt)un,t∈(tn,tn+1] (3.4) and   vΔt(t)=SCL(t−tn)un,t∈[tn,tn+1), (3.5) respectively; cf. Fig. 1. As $${u_{{{{\it {\Delta}} t}}}}$$ is discontinuous at $$t_n,$$ we introduce the right limit $${u_{{{{\it {\Delta}} t}}}}((t_n)+) = \mathcal{S}_{\text{CL}}({{{\it {\Delta}} t}})u^n$$. Similarly, let $${v_{{{{\it {\Delta}} t}}}}((t_{n+1})-) = \mathcal{S}_{\text{CL}}({{{\it {\Delta}} t}})u^n$$. 4. A priori estimates To establish the convergence of $$\left\{{{u_{{{{\it {\Delta}} t}}}}}\right\}_{{{{\it {\Delta}} t}}>0}, \left\{{{v_{{{{\it {\Delta}} t}}}}}\right\}_{{{{\it {\Delta}} t}} > 0}$$ we will need a series of a priori estimates. These are also crucial when deriving the error estimate. The following result explains the introduction of the weight functions $$\mathfrak{N}$$. Proposition 4.1 (Local $$L^p$$ estimates). Suppose (𝓐f) and (𝓐σ) are satisfied, $$2 \le p < \infty$$ and $$M \ge \left\Vert{f}\right\Vert_{\mathrm{Lip}}$$. Let $$\left\{{u^n}\right\}$$ be the splitting solutions defined by $$(3.3)$$, with initial condition $$u^0 \in L^p({\it {\Omega}},{\mathscr{F}}_0,P;L^p_\mathrm{loc}({\mathbb{R}}^d))$$. For $$t\in (0,T)$$ and $$R > 0$$, set $${\it {\Gamma}}(t) = \max\{0,R-Mt\}$$. Suppose $$\phi \in C^1({\mathbb{R}})$$ is non-negative and satisfies $$\left|{\nabla \phi}\right| \le C_\phi\phi$$. Then there exist constants $$C_1$$ and $$C_2$$ depending only on $$p,\sigma,f,C_\phi$$ such that   E[∫B(0,Γ(tn))|un(x)|pϕ(x)dx] ≤eC1tnE[∫B(0,R)|u0(x)|pϕ(x)dx] +C2tneC1tn∫B(0,R)ϕ(x)dx. (4.1) If $$\sigma(x,0) = 0$$ then $$C_2 = 0$$. Here, $$B(0,R)$$ denotes the open ball with radius $$R$$ centered at $$0$$. Remark 4.2 Suppose $$\phi \in \mathfrak{N}$$ and $$u^0 \in L^p({\it {\Omega}};L^p({\mathbb{R}}^d,\phi))$$. Then, $$\phi \in L^1({\mathbb{R}}^d)$$ and the right-hand side of (4.1) is bounded independently of $$R > 0$$. It follows that $$u^n \in L^p({\it {\Omega}};L^p({\mathbb{R}}^d,\phi))$$. Proof. 1. Deterministic step. We want to prove the following: with $$1 \le p < \infty$$, let $$v^0 \in L^p_{\mathrm{loc}}({\mathbb{R}}^d)$$ and $$v(t) = {\mathcal{S}_{\text{CL}}}(t)v^0$$. Then, for any $$0 < \tau \leq T$$,   ∫B(0,Γ(τ))|v(τ,x)|pϕ(x)dx≤e‖f‖LipCϕt∫B(0,R)|v0(x)|pϕ(x)dx. (4.2) We might as well assume $${\it {\Gamma}}(\tau) > 0$$. As $$v$$ is an entropy solution of (3.1),   ∬ΠT∫RdS(v(t,x))∂tφ+Q(v(t,x))⋅∇xφdxdt+∫RS(v0(x))φ(0,x)dx≥0 (4.3) for all non-negative $$\varphi \in C^\infty_c([0,T) \times {\mathbb{R}}^d)$$, for any convex $$S\in C^2$$ with $$S'$$ bounded and $$Q'=S'f'$$. Let $$0 < \delta < \mathrm{min}\left\{{{\it {\Gamma}}(\tau),\frac{1}{2}\tau}\right\}$$. Take   φ(t,x)=ψδ(t)Hδ(Γ(t),|x|)ϕ(x), where   ψδ(t)=1−∫0tJδ+(τ−ζ)dζ and Hδ(L,r)=∫−δLJδ(ζ−r)dζ. If $$\phi \in C^\infty({\mathbb{R}}^d)$$ then $$\varphi$$ is a non-negative function in $$C^\infty_c([0,T) \times {\mathbb{R}}^d)$$. To see this, note that by assumption, $${\it {\Gamma}}(t) > \delta$$ for all $$t \in \mathrm{supp}(\psi_\delta) \subset [0,\tau)$$. Hence, restricted to the support of $$\psi_\delta$$, $${\it {\Gamma}}(t) = R-Mt$$. Furthermore, $$H_\delta({\it {\Gamma}}(t),\left|{x}\right|) = 1$$ for all $$x \in B(0,{\it {\Gamma}}(t)-\delta)$$. By approximation, it suffices with $$\phi \in C^1({\mathbb{R}}^d)$$ for (4.3) to hold true. Recall that $$\frac{d}{\mathrm{d}t} {\it {\Gamma}}(t) = -M$$ for all $$0 \leq t \leq \tau$$ and observe that   ∂tφ(t,x) =−Jδ+(τ−t)Hδ(Γ(t),|x|)ϕ(x)−Mψδ(t)Jδ(Γ(t)−|x|)ϕ(x),∇φ(t,x) =−ψδ(t)Jδ(Γ(t)−|x|)x|x|ϕ(x)+ψδ(t)Hδ(Γ(t),|x|)∇ϕ(x). Hence,   ∫RS(v0(x))Hδ(R,|x|)ϕ(x)dx ≥∬ΠTS(v(t,x))Jδ+(τ−t)Hδ(Γ(t),|x|)ϕ(x)dxdt +∬ΠT(Q(v(t,x))⋅x|x|+MS(v(t,x)))ψδ(t)Jδ(Γ(t)−|x|)ϕ(x)dxdt⏟T1 −∬ΠTQ(v(t,x))ψδ(t)Hδ(Γ(t),|x|)⋅∇ϕ(x)dxdt⏟T2. (4.4) Suppose $$S'(0) = S(0) = 0$$. Then,   |Q(v)|=|∫0vS′(z)f′(z)dz|≤‖f‖LipS(v). It follows, as $$M \ge \left\Vert{f}\right\Vert_\mathrm{Lip}$$, that $${\mathscr{T}}^1 \ge 0$$. Because of the assumption on $$\phi$$,   |T2|≤‖f‖LipCϕ∬ΠTS(v)ψδ(t)Hδ(R,|x|)ϕ(x)dxdt. Sending $$\delta \downarrow 0$$, inequality (4.4) then takes the form   X(τ)≤X(0)+‖f‖LipCϕ∫0τX(r)dr, where   X(t)=∫B(0,Γ(t))S(v(t,x))ϕ(x)dx. Next, apply Grönwall’s inequality. Estimate (4.2) follows upon letting $$S \rightarrow \left|{\cdot}\right|^p$$ and applying the dominated convergence theorem. 2. Stochastic step. We want to prove the following. Fix $$2 \le p < \infty$$. Suppose $$w(s) \in L^p({\it {\Omega}},{\mathscr{F}}_s,P;L^p_\mathrm{loc}({\mathbb{R}}))$$ and take $$w(t) = {\mathcal{S}_{\text{SDE}}}(t,s)w(s)$$ for $$s \leq t$$. For any $$R > 0,$$ there exist constants $$C_3$$ and $$C_2$$ depending only on $$p$$ and $$\sigma$$ such that   E[∫B|w(t,x)|pϕ(x)dx]≤eC3(t−s)(E[∫B|w(s,x)|pϕ(x)dx]+C2(t−s)∫Bϕ(x)dx). (4.5) If $$\sigma(x,0) = 0$$ then $$C_2 = 0$$. By Itô’s lemma,   dS(w)=12S′′(w)σ(x,w)2dt+S′(w)σ(x,w)dB for any $$S \in C^2$$. Without loss of generality, we can assume $$p=2,4,6,\,{\ldots}\,.$$ Taking $$S(u)=\left|{u}\right|^p$$, multiplying by $$\phi$$ and integrating over $$B = B(0,R)$$, we arrive at   E[∫B|w(t,x)|pϕ(x)dx]−E[∫B|w(s,x)|pϕ(x)dx] ≤p(p−1)2∫stE[∫Bw(r,x)p−2σ(x,w(r,x))2ϕ(x)dx]dr. Recall that $$\sigma(x,w) \le \left|{\sigma(x,0)}\right| + \left\Vert{\sigma}\right\Vert_{\mathrm{Lip}}\left|{w}\right|$$. Hence, according to assumption (𝓐σ),   T3 :=p(p−1)2E[∫Bw(r,x)p−2σ(x,w(r,x))2ϕ(x)dx] ≤p(p−1)(‖σ(⋅,0)‖∞2E[∫B|w(r,x)|p−2ϕ(x)dx]  +‖σ‖Lip2E[∫B|w(r,x)|pϕ(x)dx]). Applying Hölder’s inequlity with $$\theta = \frac{p}{p-2}$$ and $$\theta' = \frac{p}{2}$$,   ∫B(|w(r,x)|pϕ(x))1/θϕ(x)1/θ′⏟|w(r,x)|p−2ϕ(x)dx≤(∫B|w(r,x)|pϕ(x)dx)1/θ⏟A(∫Bϕ(x)dx)1/θ′⏟B. Because of Young’s inequality $$AB \leq \frac{1}{\theta}A^\theta + \frac{1}{\theta'}B^{\theta'}$$. It follows that   ∫B|w(r,x)|p−2ϕ(x)dx≤p−2p∫B|w(r,x)|pϕ(x)dx+2p∫Bϕ(x)dx. Consequently,   T3 ≤(p−1)((p−2)‖σ(⋅,0)‖∞+p‖σ‖Lip2)⏟C3E[∫B|w(r,x)|pϕ(x)dx] +2(p−1)‖σ(⋅,0)‖∞2⏟C2∫Bϕ(x)dx. It follows that   E[∫B|w(t,x)|pϕ(x)dx] ≤E[∫B|w(s,x)|pϕ(x)dx] +C3∫stE[∫B|w(r,x)|pϕ(x)dx]dr+C2(∫Bϕ(x)dx)(t−s). This inequality is of the general form   X(t)≤X(s)+∫stK(r)X(r)dr+∫stH(r)dr. (4.6) Appealing to Grönwall’s inequality,   X(t)≤exp⁡[∫stK(r)dr]X(s)+∫stexp⁡[∫rtK(u)du]H(r)dr. (4.7) Identifying $$K=C_3$$ and $$H=C_2\left\Vert{\phi}\right\Vert_{L^1(B)}$$, it follows that   E[∫B|w(t,x)|pϕ(x)dx]≤eC3(t−s)E[∫B|w(s,x)|pϕ(x)dx]+C2‖ϕ‖L1(B)∫steC3(t−r)dr. Next, observe that $$e^{C_3(t-r)} \leq e^{C_3(t-s)}$$ for all $$s \leq r \leq t$$, and so (4.5) follows. 3. Inductive step. Let $$P_n$$ be the statement that (4.1) is true, and note that $$P_0$$ is trivially true. We must show that $$P_n$$ implies $$P_{n+1}$$. By (3.3), $$u^{n+1} = {\mathcal{S}_{\text{SDE}}}(t_{n+1},t_n){\mathcal{S}_{\text{CL}}}({{{\it {\Delta}} t}})u^n$$. Recall that $${v_{{{{\it {\Delta}} t}}}}((t_{n+1})-) = {\mathcal{S}_{\text{CL}}}({{{\it {\Delta}} t}})u^n$$. By (4.2),   E[∫B(0,Γ(tn+1))|vΔt((tn+1)−,x)|pϕ(x)dx]≤e‖f‖LipCϕΔtE[∫B(0,Γ(tn))|un(x)|pϕ(x)dx]. Because $$u^{n+1} = {\mathcal{S}_{\text{SDE}}}(t_{n+1},t_n){v_{{{{\it {\Delta}} t}}}}((t_{n+1})-),$$ it follows from (4.5) that   E[∫B(0,Γ(tn+1))|un+1(x)|pϕ(x)dx] ≤eC3Δt(E[∫B(0,Γ(tn+1))|vΔt((tn+1)−,x)|pϕ(x)dx]+C2∫B(0,Γ(tn+1))ϕ(x)dxΔt). Combining the two previous estimates,   E[∫B(0,Γ(tn+1))|un+1(x)|pϕ(x)dx] ≤eC3Δt(e‖f‖LipCϕΔtE[∫B(0,Γ(tn))|un(x)|pϕ(x)dx] +C2Δt∫B(0,Γ(tn+1))ϕ(x)dx) ≤eC1Δt(E[∫B(0,Γ(tn))|un(x)|pϕ(x)dx] +C2Δt∫B(0,R)ϕ(x)dx),C1=‖f‖LipCϕ+C3. Inserting the induction hypothesis brings to an end the proof of (4.1). □ Corollary 4.3 Let $${u_{{{{\it {\Delta}} t}}}}$$ and $${v_{{{{\it {\Delta}} t}}}}$$ be defined by (3.4) and (3.5), respectively, and suppose $$u^0$$ belongs to $$L^q({\it {\Omega}},{\mathscr{F}}_0,P;L^q({\mathbb{R}}^d,\phi))$$, $$2 \le q < \infty$$, $$\phi \in \mathfrak{N}$$. Then, for each $$1 \le p \le q$$, there exists a finite constant $$C$$ independent of $${{{\it {\Delta}} t}}$$ (but dependent on $$T,p,\phi,f,\sigma,u^0$$) such that   max{E[‖uΔt(t)]‖p,ϕp,E[‖vΔt(t)]‖p,ϕp}≤C,t∈[0,T]. Proof. It suffices to prove the result for $$p = q$$. To this end, suppose $$1 \le p < q$$ and $$w \in L^q({\mathbb{R}}^d,\phi)$$. Let $$r = q/p$$, $$r' = q/(q-p)$$, so that $$\frac{1}{r}+ \frac{1}{r'} = 1$$. Take $$f = \left|{u}\right|^p\phi^{1/r}$$, $$g = \phi^{1/r'}$$ and apply Hölder’s inequality. The result is   ∫Rd|w(x)|pϕ(x)dx≤(∫Rd|w(x)|qϕ(x)dx)p/q(∫Rdϕ(x)dx)1−p/q. (4.8) Consider the case $$p = q$$. By Proposition 4.1, there exists a constant $$C > 0$$ depending only on $$q,f,\sigma,u^0,T,\phi$$ such that   E[‖un‖q,ϕq]≤C,0≤n≤N. Let $$t \in [t_n, t_{n+1})$$. By (4.2),   E[‖SCL(t−tn)un⏟vΔt(t)‖q,ϕq]≤e‖f‖LipCϕΔtE[‖un‖q,ϕq]. This finishes the proof for $${v_{{{{\it {\Delta}} t}}}}$$. For $${u_{{{{\it {\Delta}} t}}}}$$, the result follows by (4.5). □ The next result should be compared with Karlsen & Storrøsten (2017, Proposition 5.2) and Chen et al. (2012, Section 6). It can be turned into a fractional $$BV_x$$ estimate ($$L^1$$-space translation estimate) along the lines of Chen et al. (2012), but we will not need this fact here. Proposition 4.4 (Fractional $$BV_x$$ estimates). Suppose (𝓐f), (𝓐f,1), (𝓐σ) and (𝓐σ,1) are satisfied. Let $$\phi \in \mathfrak{N}$$. Suppose $$u^0 \in L^2({\it {\Omega}},{\mathscr{F}}_0,P;L^2({\mathbb{R}}^d,\phi))$$. Let $$u_{{{\it {\Delta}} t}}$$ and $${v_{{{{\it {\Delta}} t}}}}$$ be defined by (3.4) and (3.5), respectively. Then there exists a constant $$C_{T}$$, independent of $${{{\it {\Delta}} t}}$$, such that   [∬Rd×Rd|uΔt(t,x+z)−uΔt(t,x−z)|Jr(z)ϕ(x)dxdz] ≤eCϕ‖f‖Lipt[∬Rd×Rd|u0(x+z)−u0(x−z))|Jr(z)ϕ(x)dxdz]+CTrκσ for any $$t\in (0,T)$$. Here $${{\kappa_\sigma}}\in (0,1/2]$$ is defined in (𝓐σ,1). If $$\sigma(x,u) = \sigma(u)$$ then we may take $$C_{T} = 0$$. The same result holds with $${u_{{{{\it {\Delta}} t}}}}$$ replaced by $${v_{{{{\it {\Delta}} t}}}}$$. Remark 4.5 In the deterministic case or whenever $$\sigma=\sigma(u)$$ is independent of the spatial location $$x$$, we recover the usual BV bound. To this end, note that $$C_T = 0$$, apply the weight $$\phi_\rho(x)=\exp{-\rho\sqrt{1+\left|{x}\right|^2}}$$ ($$\rho > 0$$) and then send $$\rho \downarrow 0$$. Before we proceed to the proof, we fix some notation and make a few observations. Let us define $$C^2$$-approximations $$\left\{{S_\delta}\right\}_{\delta > 0}$$ of the absolute value function by asking that   Sδ′(σ)=2∫0σJδ(z)dz,Sδ(0)=0. (4.9) Then,   |r|−δ≤Sδ(r)≤|r|,|Sδ″(r)|≤2δ‖J‖∞1|r|<δ. (4.10) Given $$S_\delta$$, we define $$Q_\delta$$ by   Qδ(u,v)=∫vuSδ′(ξ−v)f′(ξ)dξ,u,v∈R. (4.11) This function satisfies   |∂u(Qδ(u,v)−Qδ(v,u))|≤‖f′′‖L∞δ (4.12) and   |Qδ(u,v)|≤‖f‖LipSδ(u−v). (4.13) Let us state two convenient identities. First, for $$h = h(\cdot,\cdot)\in L^1_{{{\mathrm{loc}}}}$$,   12d∬Rd×Rdh(x,y)ϕ(x+y2)Jr(x−y2)dxdy=∬Rd×Rdh(x~+z,x~−z)ϕ(x~)Jr(z)dx~dz. (4.14) This follows by a change of variables: $$(\tilde{x},z) = \left(\frac{x+y}{2}, \frac{x-y}{2}\right)$$, $$\mathrm{d}y = 2^d \mathrm{d}z$$. Next,   12d∫Rdϕ(x+y2)Jr(x−y2)dy=(ϕ⋆Jr)(x). (4.15) Proof of Proposition 4.4. Given $$u=u(t)=u(t,x;\omega)$$, we introduce the quantity   Dru(t):=[12d∬Rd×Rd|u(t,x)−u(t,y)|Jr(x−y2)ϕ(x+y2)dxdy]. Actually, at first we are not going to work with this quantity but rather with   Dr,δu(t):=[12d∬Rd×RdSδ(u(t,x)−u(t,y))Jr(x−y2)ϕ(x+y2)dxdy], where the regularized entropy $$S_\delta$$ is defined in (4.9). In view of (4.10) and (4.15),   |Dru(t)−Dr,δu(t)|≤‖ϕ‖L1(Rd)δ,t>0. (4.16) 1. Deterministic step. Let $$v(t,x)$$ be the unique entropy solution of (3.1). We want to prove the following claim: there exists a constant $$C_1$$ depending only on $$J$$ and $$C_\phi$$ such that for all $$0 < r \le 1$$,   Dr,δv(t)≤eCϕ‖f‖Lipt(Dr,δv(0)+C1‖f″‖∞E[‖v0‖1,ϕ]t(δr)). (4.17) Let $$Q_\delta$$ be defined in (4.11). Using the entropy inequalities and Kružkov’s method of doubling the variables, it follows in a standard way that for $$t>0$$,   12d∬Rd×RdSδ(v(t,x)−v(t,y))Jr(x−y2)ϕ(x+y2)dxdy −12d∬Rd×RdSδ(v0(x)−v0(y))Jr(x−y2)ϕ(x+y2)dxdy ≤12d∫0t∬Rd×RdQδ(v(s,x),v(s,y))⋅∇ϕ(x+y2)Jr(x−y2)dxdyds +12d∫0t∬Rd×Rd(Qδ(v(s,y),v(s,x))−Qδ(v(s,x),v(s,y)))⋅∇y(ϕ(x+y2)Jr(x−y2))dxdyds =:TCL1+TCL2. By (4.13),   |TCL1|≤Cϕ‖f‖Lip12d∫0t∬Rd×RdSδ(v(s,x)−v(s,y))Jr(x−y2)ϕ(x+y2)dxdyds. Consider $${\mathscr{T}}_{\text{CL}}^2$$. Thanks to (4.12),   |Qδ(v,u)−Qδ(u,v)|=|∫vu∂ξ(Qδ(ξ,v)−Qδ(v,ξ))dξ|≤‖f″‖∞|u−v|δ, so that   |TCL2| ≤‖‖f″‖∞‖2δ12d∫0t∬Rd×Rd|v(s,x)−v(s,y)||∇Jr(x−y2)|ϕ(x+y2)dxdyds +‖f″‖∞2δ12d∫0t∬Rd×Rd|v(s,x)−v(s,y)|Jr(x−y2)|∇ϕ(x+y2)|dxdyds =:TCL2,1+TCL2,2. Consider $${\mathscr{T}}_{\text{CL}}^{2,1}$$. Setting $$\varphi_r(z) = \left\Vert{\nabla J}\right\Vert_1^{-1}\frac{1}{r^d} \left|{\nabla J(\frac{z}{r})}\right|$$, we write   |∇Jr(x−y2)|=‖∇J‖11rφr(x−y2). By the triangle inequality and (4.15),   12d∬Rd×Rd|v(s,x)−v(s,y)||∇Jr(x−y2)|ϕ(x+y2)dxdy ≤‖∇J‖12r∫Rd|v(s,x)|(ϕ⋆φr)(x)dx=‖∇J‖12r‖v(s)‖1,ϕ⋆φr. Considering $${\mathscr{T}}_{\text{CL}}^{2,2}$$, with $$\phi \in \mathfrak{N}$$,   12d∬Rd×Rd|v(s,x)−v(s,y)|Jr(x−y2)|∇ϕ(x+y2)|dxdy ≤2Cϕ∫Rd|v(s,x)|(ϕ⋆Jr)(x)dx=2Cϕ‖v(s)‖1,ϕ⋆Jr. By Lemma A.8,   max{‖v(s)‖1,ϕ⋆φr,‖v(s)‖1,ϕ⋆Jr}≤‖v(s)‖1,ϕ(1+w1,ϕ(r)), where $$w_{1,\phi}$$ is defined in Lemma A.7. Hence,   |TCL2|≤‖f″‖∞(1+w1,ϕ(r))(∫0t‖v(s)‖1,ϕds)(‖∇J‖11r+Cϕ)δ. In view of (4.2), $$\left\Vert{v(s)}\right\Vert_{1,\phi} \le e^{\left\Vert{f}\right\Vert_{\mathrm{Lip}}C_\phi s}\left\Vert{v_0}\right\Vert_{1,\phi}$$. Summarizing,   12d∬Rd×RdSδ(v(t,x)−v(t,y))Jr(x−y2)ϕ(x+y2)dxdy −12d∬Rd×RdSδ(v0(x)−v0(y))Jr(x−y2)ϕ(x+y2)dxdy ≤Cϕ‖f‖Lip⏟K∫0t12d∬Rd×RdSδ(v(s,x)−v(s,y))Jr(x−y2)ϕ(x+y2)dxdyds +∫0tC1‖f″‖∞‖v0‖1,ϕe‖f‖LipCϕs(δr)⏟H(s)ds, (4.18) where $$C_1 = (1+w_{1,\phi}(1))(\left\Vert{\nabla J}\right\Vert_1 + C_\phi) $$. This inequality is of the form (4.6). By Grönwall’s inequality (4.7),    ∬Rd×RdSδ(v(t,x)−v(t,y))Jr(x−y2)ϕ(x+y2)dxdy ≤eCϕ‖f‖Lipt(  ∬Rd×RdSδ(v0(x)−v0(y))Jr(x−y2)ϕ(x+y2)dxdy  +C1‖f″‖∞‖v0‖1,ϕt(δr)∬Rd×Rd). This proves the claim (4.17). 2. Stochastic step. Let $$w(t)={\mathcal{S}_{\text{SDE}}}(t,s)w(s)$$. We will now derive an estimate for $$w$$ similar to (4.18). There exist constants $$C_1$$ and $$C_2$$, depending only on $$J,\sigma,\phi$$, such that   Dr,δw(t)≤Dr,δw(s)+C1r2κ+1δ∫stE[‖1+|w(τ)|‖2,ϕ2]dτ+C2(t−s)δ, (4.19) for all $$0 \le r \le 1$$. If $$M_{\sigma} = 0$$ then $$C_1 = 0$$. Since $$w(t,x)-w(t,y)$$ solves   d(w(t,x)−w(t,y))=(σ(x,w(t,x))−σ(y,w(t,y)))dB(t) applying Itô’s formula to $$S_\delta(w(t,x)-w(t,y))$$ yields   dSδ(w(t,x)−w(t,y)) =12Sδ′′(w(t,x)−w(t,y))(σ(x,w(t,x))−σ(y,w(t,y)))2dt, +Sδ′(w(t,x)−w(t,y))(σ(x,w(t,x))−σ(y,w(t,y)))dB(t). Integrating against the test function $$\frac{1}{2^d}J_r(\tfrac{x-y}{2})\phi(\tfrac{x+y}{2})$$, we arrive at   12d∬Rd×RdSδ(w(t,x)−w(t,y))Jr(x−y2)ϕ(x+y2)dxdy −12d∬Rd×RdSδ(w(s,x)−w(s,y))Jr(x−y2)ϕ(x+y2)dxdy =∫st12d∬Rd×Rd12Sδ′′(w(τ,x)−w(τ,y)) ×(σ(x,w(τ,x))−σ(y,w(τ,y)))2Jr(x−y2)ϕ(x+y2)dxdydτ +∫st12d∬Rd×RdSδ′(w(τ,x)−w(τ,y))(σ(x,w(τ,x))−σ(y,w(τ,y)))dxdydB(τ) =TSDE1+TSDE2, where the $${\mathscr{T}}_{\text{SDE}}^2$$ term has zero expectation. Note that   (σ(x,u)−σ(y,v))2≤2(σ(x,u)−σ(x,v))2+2(σ(x,v)−σ(y,v))2 for any $$u,v \in {\mathbb{R}}$$. We estimate the $${\mathscr{T}}_{\text{SDE}}^1$$ term as follows:   E[|TSDE1|] ≤2E[∫st12d∬Rd×RdJδ(w(τ,x)−w(τ,y))(σ(x,w(τ,x))−σ(y,w(τ,x)))2  ×Jr(x−y2)ϕ(x+y2)dxdydτ\raisebox25pt ] +2E[∫st12d∬Rd×RdJδ(w(τ,x)−w(τ,y))(σ(y,w(τ,x))−σ(y,w(τ,y)))2  ×Jr(x−y2)ϕ(x+y2)dxdydτ\raisebox25pt ]=:S1+S2. Regarding $$S_1$$, recall that $$\left|{J_\delta}\right| \le \left\Vert{J}\right\Vert_\infty/\delta$$. By (𝓐σ,1),   |S1| ≤‖J‖∞2δE[∫st12d∬Rd×Rd|σ(x,w(τ,x))−σ(y,w(τ,x))|2×Jr(x−y2)ϕ(x+y2)dxdydτ] ≤‖J‖∞Mσ22δE[∫st12d∬Rd×Rd|x−y|2κσ+1(1+|w(τ,x)|)2×Jr(x−y2)ϕ(x+y2)dxdydτ] ≤2‖J‖∞Mσ2(2r)2κσ+1δ∫stE[‖1+|w(τ)|‖2,ϕ⋆Jr2]dτ. By Lemma A.8,   ‖1+|w(τ)|‖2,ϕ⋆Jr2≤‖1+|w(τ)|‖2,ϕ2(1+w1,ϕ(r)), where $$w_{1,\phi}$$ is defined in Lemma A.7. It follows that   |S1|≤22(κσ+1)‖J‖∞Mσ2(1+w1,ϕ(1))⏟C1∫stE[‖1+|w(τ)|‖2,ϕ2]dτr2κσ+1δ for all $$0 < r \leq 1$$. Consider $$S_2$$. Because of assumption (𝓐σ),   Jδ(w(τ,x)−w(τ,y))(σ(y,w(τ,x))−σ(y,w(τ,y)))2≤‖σ‖Lip2‖J‖∞δ. Hence,   |S2|≤2‖σ‖Lip2‖J‖∞‖ϕ‖L1(Rd)⏟C2(t−s)δ. This proves (4.19) 3. Inductive step. Let $$P_n$$ be the following claim: there exist constants $$C_1,C_2,C_3$$ depending only on $$J,\phi,\sigma$$ such that for all $$0 < r \le 1$$,   Dr,δun ≤eCϕ‖f‖Liptn(Dr,δu0+C3‖f″‖∞(Δt∑k=0n−1E[‖uk‖1,ϕ])δr  +C1r2κσ+1δ∫0tnE[‖1+|uΔt(t)|‖2,ϕ2]dt+C2tnδ). (4.20) If $$M_{\sigma} = 0$$ then $$C_1 = 0$$. Note that $$P_0$$ is trivially true. Assuming that $$P_n$$ is true, we want to verify $$P_{n+1}$$. Recall that $$u^{n+1} = {\mathcal{S}_{\text{SDE}}}(t_{n+1},t_n){\mathcal{S}_{\text{CL}}}({{{\it {\Delta}} t}})u^n$$. Let $$w^n = {\mathcal{S}_{\text{CL}}}({{{\it {\Delta}} t}})u^n$$ and note that $${\mathcal{S}_{\text{SDE}}}(t,t_n)w^n = u_{{{\it {\Delta}} t}}(t)$$ for $$t_n \le t <t_{n+1}$$. As $$1 \le e^{C_\phi\left\Vert{f}\right\Vert_{\mathrm{Lip}}{{{\it {\Delta}} t}}}$$, (4.19) yields   Dr,δun+1≤Dr,δwn+eCϕ‖f‖LipΔt(C1r2κσ+1δ∫tntn+1E[‖1+|uΔt(t)|‖2,ϕ2]dt+C2Δtδ). By (4.17),   Dr,δwn≤eCϕ‖f‖LipΔt(Dr,δun+C3‖f″‖∞E[‖un‖1,ϕ]Δt(δr)). Hence,   Dr,δun+1 ≤eCϕ‖f‖LipΔt(Dr,δun+C3‖f″‖∞E[‖un‖1,ϕ]Δt(δr) +C1r2κσ+1δ∫tntn+1E[‖1+|uΔt(t)|‖2,ϕ2]dt+C2Δtδ) and inserting the hypothesis $$P_n$$ yields $$P_{n+1}$$. 4. Concluding the proof. Consider (4.20). By Corollary 4.3, there exists a constant $$C$$, independent of $${{{\it {\Delta}} t}}$$, such that   Dr,δun≤eCϕ‖f‖Liptn(Dr,δu0+Ctn(δr+δ+r2κσ+1δ)). Because of (4.16), this translates into   Drun≤eCϕ‖f‖Liptn(Dru0+Ctn(δr+δ+r2κσ+1δ)+2‖ϕ‖L1(Rd)δ),0≤n≤N. We can argue via (4.19) to obtain   DruΔt(t)≤eCϕ‖f‖Lipt(Dru0+Ct(δr+δ+r2κσ+1δ)+2‖ϕ‖L1(Rd)δ),t∈[0,T]. Note that the same holds true if we replace $${u_{{{{\it {\Delta}} t}}}}$$ by $${v_{{{{\it {\Delta}} t}}}}$$, thanks to (4.17). Viewing $$r > 0$$ as fixed, we can choose $$\delta = r^{{{\kappa_\sigma}} +1}$$ to arrive at the bound   DruΔt(t)≤eCϕ‖f‖LiptDru0+CTrκσ. The result follows by (4.14). In the case that $$M_{\sigma} = 0$$, we have   DruΔt(t)≤eeCϕ‖f‖LiptDru0+CT(δr+δ),t∈(0,T), and we may send $$\delta \downarrow 0$$ independently of $$r$$. □ In Proposition 4.4, the spatial regularity of $${u_{{{{\it {\Delta}} t}}}}, {v_{{{{\it {\Delta}} t}}}}$$ is characterized in terms of averaged $$L^1$$-space translates. In the BV context, this is equivalently characterized by integration against the divergence of a smooth bounded function. Restricting to one dimension ($$d = 1$$) and $$u \in C^1({\mathbb{R}})$$, we have   suph>0{1h∫R|u(x+h)−u(x)|dx}=∫R|u′(x)|dx=sup{∫Ru(x)β′(x)dx:β∈Cc∞(R),‖β‖∞≤1}. Fix $$\kappa \in (0,1]$$. The left-hand side has a natural generalization to the fractional BV setting by considering $$u \in L^1({\mathbb{R}})$$ satisfying   suph>0{1hκ∫R|u(x+h)−u(x)|dx}<∞. (4.21) A possible generalization of the right-hand side reads   sup{δ1−κ∫Ru(x)(Jδ⋆β)′(x)dx:δ>0,‖β‖∞≤1}<∞, (4.22) where $$\left\{{J_\delta}\right\}_{\delta > 0}$$ is a suitable family of symmetric mollifiers. The next lemma shows that (4.22) may be bounded in terms of (4.21). The lemma plays a key role in obtaining the optimal $$L^1$$-time continuity estimates in Proposition 4.8. Lemma 4.6 Let $$\rho \in C^\infty_c((0,1))$$ satisfy $$\int_0^1 \rho(r)\,\mathrm{d}r = 1$$ and $$\rho \geq 0$$. For $$x \in {\mathbb{R}}^d$$ define   U(x)=1α(d)Md(1−∫0|x|ρ(r)dr),V(x)=1dα(d)Md−1ρ(|x|), where $$M_n = \int_0^\infty r^n\rho(r)\,\mathrm{d}r$$, $$n \geq 0$$ and $$\alpha(d)$$ denotes the volume of the unit ball in $${\mathbb{R}}^d$$. Then $$U,V$$ are symmetric mollifiers on $${\mathbb{R}}^d$$ with support in $$B(0,1)$$. For $$\phi \in \mathfrak{N}$$, $$u \in L^1({\mathbb{R}}^d,\phi)$$ and $$\delta > 0$$, define   Vδ(u)=∬Rd×Rd|u(x+z)−u(x−z)|Vδ(z)ϕ(x)dzdx, where $$V_\delta(z) = \delta^{-d}V(\delta^{-1}z)$$. Similarly, for $$\beta \in L^\infty({\mathbb{R}}^d)$$ let   Uδi(u,β)=∫Rdu(x)∂xi(Uδ⋆β)(x)ϕ(x)dx,1≤i≤d, where $$U_\delta(z) = \delta^{-d}U(\delta^{-1}z)$$. Then   |Uδi(u,β)|≤dMd−12Md(1δVδ(u)+2‖u‖1,ϕw1,ϕ(δ)δ)‖β‖L∞, for each $$1 \leq i \leq d$$, where $$w_{1,\phi}$$ is defined in Lemma A.7. Remark 4.7 We note that Lemma 4.6 covers the BV case. If there is a constant $$C \geq 0$$ such that $$\mathcal{V}_\delta(u) \leq C \delta$$ (the BV case) then   ∫Rdu(∇⋅β)ϕdx=limδ↓0∑i=1dUδi(u,βi)≤d2Md−12Md(C+2Cϕ‖u‖1,ϕ) for any $$\beta = (\beta^1, \dots,\beta^d) \in C^1_c({\mathbb{R}}^d;{\mathbb{R}}^d)$$ satisfying $$\left\Vert{\beta}\right\Vert_\infty \leq 1$$. It follows that   ∫Rd|∇u|ϕdx ≤sup|β|≤1∫Rd(∇u⋅β)ϕdx =sup|β|≤1∫Rdu(∇⋅β)ϕ+u(β⋅∇ϕ)dx ≤d2Md−12Md(C+2Cϕ‖u‖1,ϕ)+Cϕ‖u‖1,ϕ and so $$\left|{\nabla u}\right|$$ is a finite measure with respect to $$\phi\, \mathrm{d}x$$. Proof. Let us first show that $$U$$ is a symmetric mollifier. It is clearly symmetric, furthermore it is smooth since $$\left\{{0}\right\} \notin \mathrm{cl}(\mathrm{supp}(\rho))$$. Change to polar coordinates and integrate by parts to obtain   ∫Rd(1−∫0|x|ρ(σ)dσ)dx =α(d)∫0∞drd−1(1−∫0rρ(σ)dσ)dr =α(d)∫0∞rdρ(r)dr=α(d)Md. Similarly for $$V$$,   ∫Rdρ(|x|)dx=dα(d)∫0∞rd−1ρ(r)dr=dα(d)Md−1. Note that   Uδi(u,β)=∬Rd×Rdu(x)∂xiUδ(x−y)β(y)ϕ(x)dydx. Next, we differentiate to obtain   ∂xiUδ(x)=−1α(d)Md1δdρ(|x|δ)1δsign(xi)=−dMd−1MdVδ(x)1δsign(xi). Hence,   Uδi(u,β)=−dMd−1Md1δ∬Rd×Rdu(x)Vδ(x−y)sign(xi−yi)β(y)dydx. This integral may be reformulated according to    ∬Rd×Rdu(x)Vδ(x−y)sign(xi−yi)β(y)ϕ(x)dydx =12∬Rd×Rdu(x)Vδ(x−y)sign(xi−yi)β(y)ϕ(x)dydx −12∬Rd×Rdu(x)Vδ(y−x)sign(yi−xi)β(y)ϕ(x)dydx =12∬Rd×Rd(u(y−z)ϕ(y−z)−u(y+z)ϕ(y+z))Vδ(z)sign(zi)β(y)dzdy, where we made the substitution $$x = y-z$$ and $$x = y + z,$$ respectively. Since   u(y−z)ϕ(y−z)−u(y+z)ϕ(y+z) =(u(y−z)−u(y+z))ϕ(y) +u(y−z)(ϕ(y−z)−ϕ(y))−u(y+z)(ϕ(y+z)−ϕ(y)), it follows that   Uδi(u,β) =dMd−12Md1δ∬Rd×Rd(u(y+z)−u(y−z))Vδ(z)sign(zi)β(y)ϕ(y)dzdy +dMd−12Md1δ∬Rd×Rdu(y+z)(ϕ(y+z)−ϕ(y))Vδ(z)sign(zi)β(y)dzdy +dMd−12Md1δ∬Rd×Rdu(y−z)(ϕ(y)−ϕ(y−z))Vδ(z)sign(zi)β(y)dzdy =:Zδ1+Zδ2+Zδ3. Clearly,   |Zδ1|≤dMd−12Md1δ∬Rd×Rd|u(y+z)−u(y−z)|Vδ(z)ϕ(y)dzdy⏟Vδ(u)‖β‖L∞. Consider $${\mathscr{Z}}_\delta^2$$; the term $${\mathscr{Z}}_\delta^3$$ is treated similarly. By Lemma A.7,   |ϕ(y+z)−ϕ(y)|≤w1,ϕ(|z|)ϕ(y+z). Hence, by Young’s inequality for convolutions,   |Zδ2|≤dMd−12Mdw1,ϕ(δ)δ∫Rd(|uϕ|⋆Vδ)(y)dy‖β‖L∞≤dMd−12Mdw1,ϕ(δ)δ‖u‖1,ϕ‖β‖L∞. This concludes the proof of the lemma. □ Next, we consider the time continuity of the splitting approximations. Recall that the interpolants $${u_{{{{\it {\Delta}} t}}}},{v_{{{{\it {\Delta}} t}}}}$$ are discontinuous at $$t_n = n{{{\it {\Delta}} t}}$$. Hence, the result must somehow quantify the size of the jumps as $${{{\it {\Delta}} t}} \downarrow 0$$. The idea of the proof is to ‘transfer à la Kružkov’ spatial regularity to temporal continuity Kružkov (1969, 1970). Given a bounded variation bound, or some spatial $$L^1$$-modulus of continuity, this approach has been applied to miscellaneous splitting methods for deterministic problems; cf. Holden et al. (2010) (and references therein). At variance with Holden et al. (2010), we quantify spatial regularity differently, namely in terms of averaged (weighted) $$L^1$$-translates. Combined with Lemma 4.6, we deduce $$L^1$$-time continuity estimates that recover the optimal estimates in the $$BV_x$$ case ($$\kappa=1$$). Proposition 4.8 ($$L^1$$-time continuity). Assume that (𝓐f), (𝓐f,1), (𝓐σ) and (𝓐σ,1) hold. Fix $$\phi \in \mathfrak{N}$$, and let $$u^0 \in L^2({\it {\Omega}},{\mathscr{F}}_0,P;L^2({\mathbb{R}}^d,\phi))$$ satisfy   E[∬Rd×Rd|u0(x+z)−u0(x−z))|Jr(z)ϕ(x)dxdz]=O(rκ0) (4.23) for any symmetric mollifier $$J$$ and some $$0 < {{\kappa_0}} \leq 1$$. Set   κ:={min{κ0,κσ} if σ=σ(x,u),κ0 if σ=σ(u).  Let $${u_{{{{\it {\Delta}} t}}}}$$ and $${v_{{{{\it {\Delta}} t}}}}$$ be defined in (3.4) and (3.5), respectively. (i) Suppose $$0 < \tau_1 < \tau_2 \le T$$ satisfy $$\tau_1\in (t_k,t_{k+1}]$$ and $$\tau_2\in (t_l,t_{l+1}]$$. Then there exists a finite constant $$C_{T,\phi}$$, independent of $${{{\it {\Delta}} t}}$$, such that   E[∫Rd|uΔt(τ2,x)−uΔt(τ1,x)|ϕ(x)dx]≤CT,ϕ(|(l−k)Δt|κ+τ2−τ1). (ii) Suppose $$0 \le \tau_1 \le \tau_2 < T$$ satisfy $$\tau_1\in [t_k,t_{k+1})$$ and $$\tau_2\in [t_l,t_{l+1})$$. Then there exists a finite constant $$C_{T,\phi}$$, independent of $${{{\it {\Delta}} t}}$$, such that   E[∫Rd|vΔt(τ2,x)−vΔt(τ1,x)|ϕ(x)dx]≤CT,ϕ((l−k)Δt+|τ2−τ1|κ). Proof. We shall first quantify weak continuity in the mean of $$t\mapsto {u_{{{{\it {\Delta}} t}}}}(t)$$, $$t\mapsto {v_{{{{\it {\Delta}} t}}}}(t)$$ and then turn this into fractional $$L^1$$-time continuity in the mean. The reason for first exhibiting a weak estimate is that the splitting steps do not produce functions that are Lipschitz continuous in time, thereby preventing a direct ‘inductive argument’ (see Kružkov, 1969). 1. Weak estimate. Let $$t_n = n {{{\it {\Delta}} t}}$$. Suppose $$0 < \tau_1 \le \tau_2 \le T$$ satisfies $$\tau_1\in (t_k,t_{k+1}]$$ and $$\tau_2\in (t_l,t_{l+1}]$$. Suppose $$\beta$$ belongs to $$L^\infty({\it {\Omega}} \times {\mathbb{R}}^d,{\mathscr{F}} \otimes {\mathscr{B}\left({{\mathbb{R}}^d}\right)},{\rm d}P \otimes \mathrm{d}x)$$ and let $$\beta_\delta = \beta \star U_\delta$$, where $$U_\delta$$ is defined in Lemma 4.6. We claim that there is a constant $$C > 0$$, independent of $${{{\it {\Delta}} t}}$$, such that   E[∫Rd(uΔt(τ2,x)−uΔt(τ1,x))(βδϕ)(x)dx]≤C(δκ−1(l−k)Δt+τ2−τ1)‖β‖L∞. (4.24) Consider the case $$l \geq k+1$$. We continue as follows:   T =E[∫Rd(uΔt(τ2,x)−uΔt(τ1,x))(βδϕ)(x)dx] =E[∫Rd(uΔt(τ2,x)−uΔt((tl)+,x))(βδϕ)(x)dx] +E[∫Rd(uΔt(tk+1,x)−uΔt(τ1,x))(βδϕ)(x)dx] +E[∫Rd(uΔt((tl)+,x)−uΔt(tk+1,x))(βδϕ)(x)dx] =:T1+T2+T3. Recall that $${u_{{{{\it {\Delta}} t}}}}((t_n)+) = {v_{{{{\it {\Delta}} t}}}}((t_{n+1})-) = {\mathcal{S}_{\text{CL}}}({{{\it {\Delta}} t}})u^n$$. Regarding the last term,   uΔt((tl)+,x)−uΔt(tk+1,x)=vΔt((tl+1)−,x)−vΔt(tl,x)+∑n=k+1l−1uΔt(tn+1,x)−uΔt(tn,x), where the sum is empty for the case $$l = k+1$$. Furthermore, we note that   uΔt(tn+1,x)−uΔt(tn,x)=(uΔt(tn+1,x)−uΔt((tn)+,x))+(vΔt((tn+1)−,x)−vΔt(tn,x)). This yields   T3 =E[∑n=k+1l−1∫Rd(uΔt(tn+1,x)−uΔt((tn)+,x))(βδϕ)(x)dx] +E[∑n=k+1l∫Rd(vΔt((tn+1)−,x)−vΔt(tn,x))(βδϕ)(x)dx] =E[∫Rd(∫tk+1tlσ(x,uΔt(t,x))dB(t))(βδϕ)(x)dx] +E[∑n=k+1l∫Rd(SCL(Δt)un(x)−un(x))(βδϕ)(x)dx]. It follows that $${\mathscr{T}} = {\mathscr{T}}^{\text{CL}} + {\mathscr{T}}^{\text{SDE}}$$, where   TCL :=E[∑n=k+1l∫Rd(SCL(Δt)un(x)−un(x))(βδϕ)(x)dx],TSDE :=E[∫Rd∫τ1τ2σ(x,uΔt(t,x))dB(t)(βδϕ)(x)dx]. Note that this holds true for $$k = l$$ as $${\mathscr{T}}^{\text{CL}} = 0$$ in this case. As $${v_{{{{\it {\Delta}} t}}}}(t,x)$$ is a weak solution of the conservation law (3.1) on $$[t_n,t_{n+1})$$,   |∫Rd(vΔt((tn+1)−,x)−vΔt(tn,x))(βδϕ)(x)dx| ≤|∫tntn+1∫Rdf(vΔt(r,x))⋅∇(βδϕ)(x)dxdr| ≤|∫tntn+1∫Rdf(vΔt(r,x))⋅∇βδ(x)ϕ(x)dxdr| +|∫tntn+1∫Rdf(vΔt(r,x))⋅(βδ(x)∇ϕ(x))dxdr| =:Zδ1+Zδ2. By Proposition 4.4, there exists a constant $$C > 0$$ such that   E[∬Rd×Rd|vΔt(r,x+z)−vΔt(r,x−z)]|Vδ(z)ϕ(x)dzdx≤Cδκ. Consequently, taking expectations in Lemma 4.6 yields   E[Zδ1≤d2Md−12Md‖f‖Lip(ΔtCδκ−1+2E[∫tntn+1‖vΔt(r)‖1,ϕdr]]δ−1w1,ϕ(2δ))‖β‖L∞. As $$\phi \in \mathfrak{N}$$,   Zδ2≤‖f‖LipCϕE[∫tk+1tl+1‖vΔt(t)]‖1,ϕdt‖β‖L∞. Summarizing, there exists a constant $$C$$ such that   |TCL|≤Cδκ−1(l−k)Δt‖β‖L∞ for all $$0 < \delta \le 1$$. By (4.8), Jensen’s inequality and the Itô isometry,   |TSDE| ≤‖β‖L∞∫RdE[|∫τ1τ2σ(x,uΔt(t,x))dB(t)|]ϕ(x)dx ≤‖β‖L∞‖ϕ‖L1(Rd)1/2(∫RdE[|∫τ1τ2σ(x,uΔt(t,x))dB(t)|]2ϕ(x)dx)1/2 ≤‖β‖L∞‖ϕ‖L1(Rd)1/2(∫RdE[∫τ1τ2σ2(x,uΔt(t,x))dt]ϕ(x)dx)1/2 =‖β‖L∞‖ϕ‖L1(Rd)1/2(∫τ1τ2E[‖σ(⋅,uΔt(t,⋅))]‖2,ϕ2dt)1/2 ≤C‖β‖L∞‖ϕ‖L1(Rd)1/2τ2−τ1, since, in view of (𝓐σ) and Corollary 4.3, $${E \left[{\left\Vert{\sigma(\cdot,{u_{{{{\it {\Delta}} t}}}}(t,\cdot))}\right]}\right\Vert^2_{2,\phi}}^{1/2} \leq C$$ for some constant $$C$$ independent of $$t \in [0,T]$$. Summarizing, the above estimates imply the existence of a constant $$C$$, independent of $${{{\it {\Delta}} t}},\delta$$ and $$\beta$$, such that   |T|≤C(δκ−1(k−l)Δt+τ2−τ1)‖β‖L∞, which yields (4.24). Let us consider $${v_{{{{\it {\Delta}} t}}}}$$. Suppose $$0 \le \tau_1 \le \tau_2 < T$$, with $$\tau_1 \in [t_k,t_{k+1})$$, $$\tau_2 \in [t_l,t_{l+1})$$. We claim there is a constant $$C > 0$$, independent of $${{{\it {\Delta}} t}},\delta$$ and $$\beta$$, such that   E[∫Rd(vΔt(τ2,x)−vΔt(τ1,x))(βδϕ)(x)dx]≤C(δκ−1|τ2−τ1|+(l−k)Δt)‖β‖L∞. (4.25) To prove this claim, note that   vΔt(τ2,x)−vΔt(τ1,x) =vΔt(τ2,x)−vΔt(tl,x) +∑n=k+1lvΔt(tn,x)−vΔt((tn)−,x) +∑n=k+1l−1vΔt((tn+1)−,x)−vΔt(tn,x) +vΔt((tk+1)−,x)−vΔt(τ1,x), and so   E[∫Rd(vΔt(τ2,x)−vΔt(τ1,x))(βδϕ)(x)dx]=TCL+TSDE, where   TCL :=E[∫Rd(SCL(τ2−tl)ul(x)−ul(x))(βδϕ)(x)dx] +∑n=k+1l−1E[∫Rd(SCL(Δt)un(x)−un(x))(βδϕ)(x)dx] +E[∫Rd(SCL(Δt)uk−SCL(τ1−tk)uk)(βδϕ)(x)dx],TSDE :=∑n=k+1lE[∫Rd(∫tn−1tnσ(x,uΔt(t,x))dB(t))(βδϕ)(x)dx] =E[∫Rd(∫tktlσ(x,uΔt(t,x))dB(t))(βδϕ)(x)dx]. Combining the above estimates yields (4.25). 2. Strong estimate. Let $$d(x)= {u_{{{{\it {\Delta}} t}}}}(\tau_2)-{u_{{{{\it {\Delta}} t}}}}(\tau_1)$$, $$\beta(x) = {\mathrm{sign}\left({d(x)}\right)}$$. By the triangle inequality,   E[∫Rd|uΔt(τ2,x)−uΔt(τ1,x)|ϕ(x)dx] ≤|E[∫Rdβδ(x)d(x)ϕ(x)dx]|+E[∫Rd||d(x)|−βδ(x)d(x)|ϕ(x)dx] =:T1+T2. By (4.24),   T1=O(δκ−1(l−k)Δt+τ2−τ1). Consider $${\mathscr{T}}_2$$. Following, e.g., Kružkov (1970, Lemma 1),   ||d(x)|−βδ(x)d(x)|≤∫Rd||d(x)|−d(x)sign(d(y))|Vδ(x−y)dy ≤2∫Rd|d(x)−d(y)|Vδ(x−y)dy. Upon adding and subtracting identical terms and changing variables $$2\tilde x = x+y$$, $$2z = x-y$$, it follows (after relabeling $$\tilde x$$ by $$x$$)   ∫Rd||d(x)|−βδ(x)d(x)|ϕ(x)dx ≤2∬Rd×Rd|d(x+z)−d(x−z)| ×Vδ/2(z)|ϕ(x+z)−ϕ(x)|dzdx +2∬Rd×Rd|d(x+z)−d(x−z)|Vδ/2(z)ϕ(x)dzdx =:T21+T22. Consider $${\mathscr{T}}_2^1$$. By Lemma A.7,   |ϕ(x+z)−ϕ(x)|≤w1,ϕ(|z|)ϕ(x). Hence, by the symmetry of $$V$$ and the triangle inequality,   |T21| ≤4∬Rd×Rd|d(x−z)|Vδ/2(z)w1,ϕ(|z|)ϕ(x)dzdx ≤4w1,ϕ(δ/2)∬Rd×Rd|d(y)|Vδ/2(x−y)ϕ(x)dydx ≤2w1,ϕ(δ)‖uΔt(τ2)−uΔt(τ1)‖1,ϕ⋆Vδ/2. By Lemma A.8 and Corollary 4.3, $${E \left[{\left|{{\mathscr{T}}_2^1}\right]}\right|} = {\mathcal{O}}(\delta)$$. By Proposition 4.4, it follows in view of assumption (4.23) that $${E \left[{\left|{{\mathscr{T}}_2^2}\right]}\right|} = {\mathcal{O}}(\delta^\kappa)$$. Consequently,   T1+T2=O(δκ−1(l−k)Δt+τ2−τ1+δκ). Choosing $$\delta=((l-k){{{\it {\Delta}} t}})$$ concludes the proof of (i). Result (ii) follows analoguously due to (4.25). □ 5. Convergence Equipped with $${{{\it {\Delta}} t}}$$-uniform a priori estimates, we are now prepared to study the limiting behavior of $${u_{{{{\it {\Delta}} t}}}}, {v_{{{{\it {\Delta}} t}}}}$$ as $${{{\it {\Delta}} t}}\downarrow 0$$. As discussed in Section 1, we will apply the framework of Young measures. We refer to the appendix (Section A.5) for some background material on Young measures and weak compactness. We start by establishing an approximate entropy inequality for the operator splitting solutions. Lemma 5.1 Suppose $$u^0 \in L^2({\it {\Omega}},{\mathscr{F}}_0,P;L^2({\mathbb{R}}^d,\phi))$$, $$\phi \in \mathfrak{N}$$. Let $${u_{{{{\it {\Delta}} t}}}}$$ and $${v_{{{{\it {\Delta}} t}}}}$$ be defined by (3.4) and (3.5), respectively. For any $$(S,Q) \in \mathscr{E}$$, any $$V \in {\mathcal{S}}$$ and any non-negative $${\varphi} \in C^\infty_c([0,T) \times {\mathbb{R}}^d)$$,   0 ≤E[∫RdS(u0(x)−V)φ(0,x)dx] +E[∬ΠTS(uΔt(t,x)−V)∂tφ(t,x)+Q(vΔt(t,x),V)⋅∇φ(t,x)dxdt] −E[∬ΠTS″(uΔt(t,x)−V)DtVσ(x,uΔt(t,x))φ(t,x)dxdt] +E[12∬ΠTS″(uΔt(t,x)−V)σ2(x,uΔt(t,x))φ(t,x)dxdt] +∑n=0N−1E[∫tntn+1∫Rd(S(vΔt(t,x)−V)−S(vΔt((tn+1)−,x)−V)∂tφ(t,x)dxdt]. (5.1) Proof. Let us for the moment assume that $$u^0 \in L^p({\it {\Omega}},{\mathscr{F}}_0,P;L^p({\mathbb{R}}^d,\phi))$$ for all $$2 \leq p < \infty$$. By definition, $${v_{{{{\it {\Delta}} t}}}}$$ satisfies    ∫RdS(un(x)−V)φ(tn,x)dx−∫RdS(vΔt((tn+1)−,x)−V)φ(tn+1,x)dx +∫tntn+1∫RdS(vΔt(t,x)−V)∂tφ(t,x)dxdt +∫tntn+1∫RdQ(vΔt(t,x),V)⋅∇φ(t,x)dxdt≥0. For fixed $$x \in {\mathbb{R}}^d$$, apply Theorem A.10 with $$F(\zeta,\lambda,t) = S(\zeta-\lambda)\varphi(t,x)$$ and   uΔt(t,x)⏟X(t)=uΔt((tn)+,x)⏟X0+∫tntσ(x,uΔt(s,x))⏟u(s)dB(s). This yields, after integrating in space,   ∫RdS(un+1(x)−V)φ(tn+1,x)dx =∫RdS(uΔt((tn)+)−V)φ(tn,x)dx +∫Rd∫tntn+1S(uΔt(t,x)−V)∂tφ(t,x)dtdx +∫Rd∫tntn+1S′(uΔt(t,x)−V)σ(x,uΔt(t,x))φ(t,x)dB(t)dx −∫Rd∫tntn+1S″(uΔt(t,x)−V)DtVσ(x,uΔt(t,x))φ(t,x)dtdx +12∫Rd∫tntn+1S″(uΔt(t,x)−V)σ2(x,uΔt(t,x))φ(t,x)dtdx, where the stochastic integral is a Skorohod integral. Note that    ∫RdS(uΔt((tn)+,x)−V)φ(tn,x)dx−∫RdS(vΔt((tn+1)−,x)−V)φ(tn+1,x)dx =−∫Rd∫tntn+1S(vΔt((tn+1)−,x)−V)∂tφ(t,x)dtdx. Adding the two equations and taking expectations we attain   E[∫RdS(un(x)−V)φ(tn,x)dx−E[∫RdS(un+1(x)−V)φ(tn+1,x)dx]] +E[∫tntn+1∫Rd(S(vΔt(t,x)−V)−S(vΔt((tn+1)−,x)−V))∂tφ(t,x)dxdt] +E[∫Rd∫tntn+1S(uΔt(t,x)−V)∂tφ(t,x)dtdx] +E[∫tntn+1∫RdQ(vΔt(t,x),V)⋅∇φ(t,x)dxdt] −E[∫Rd∫tntn+1S″(uΔt(t,x)−V)DtVσ(x,uΔt(t,x))φ(t,x)dtdx] +E[12∫Rd∫tntn+1S″(uΔt(t,x)−V)σ2(x,uΔt(t,x))φ(t,x)dtdx]≥0, where we applied the fact that the Skorohod integral has zero expectation. Next we sum over $$n = 0,1,\dots,N-1$$. This yields (5.1). The result follows for general $$u^0 \in L^2({\it {\Omega}},{\mathscr{F}}_0,P;L^2({\mathbb{R}}^d,\phi))$$ by approximation. □ Theorem 5.2 Suppose (𝓐f), (𝓐f,1), (𝓐σ) and (𝓐σ,1) hold. Let $$\phi \in \mathfrak{N}$$ and $$2 \le p < \infty$$. Suppose $$u^0 \in L^p({\it {\Omega}},{\mathscr{F}}_0,P;L^p({\mathbb{R}}^d,\phi))$$ satisfies (4.23). Let $${u_{{{{\it {\Delta}} t}}}}$$ and $${v_{{{{\it {\Delta}} t}}}}$$ be defined by (3.4) and (3.5), respectively. Then there exists a subsequence $$\left\{{{{{\it {\Delta}} t}}_j}\right\}$$ and a predictable $$u \in L^p([0,T] \times {\it {\Omega}};L^p({\mathbb{R}}^d \times [0,1],\phi))$$ such that both $${u_{{{{\it {\Delta}} t}}}}j \rightarrow u$$ and $${v_{{{{\it {\Delta}} t}}}}j \rightarrow u$$ in the following sense: for any Carathéodory function $$\Psi:{\mathbb{R}} \times {\it {\Pi}}_T \times {\it {\Omega}} \rightarrow {\mathbb{R}}$$ such that $$\Psi({u_{{{{\it {\Delta}} t}}}}j,\cdot) \rightharpoonup \overline{\Psi}$$ (respectively, $$\Psi({v_{{{{\it {\Delta}} t}}}}j,\cdot) \rightharpoonup \overline{\Psi}$$) in $$L^1({\it {\Pi}}_T \times {\it {\Omega}},\phi\,\mathrm{d}x \otimes \mathrm{d}t \otimes {\rm d}P)$$,   Ψ¯(t,x,ω)=∫01Ψ(u(t,x,α,ω),t,x,ω)dα. (5.2) The process $$\tilde{u} = \int_0^1 u \,{\rm d}\alpha$$ is an entropy solution in the sense of Definition 2.1 with initial condition $$u^0$$. Proof. 1. Existence of limits. Let us investigate the limit behavior of $${u_{{{{\it {\Delta}} t}}}}$$, noting that the same considerations apply to $${v_{{{{\it {\Delta}} t}}}}$$. We argue as in Karlsen & Storrøsten (2017, Theorem 4.1, Step 1) (see also (see also Bauzet et al., 2012, Section A.3.3). We apply Theorem A.11 to $$\left\{{{u_{{{{\it {\Delta}} t}}}}}\right\}$$ on the measure space   (X,A,μ)=(Ω×ΠT,P⊗B(Rd),dP⊗dt⊗ϕdx). By Corollary 4.3,   supΔt>0{E[∬ΠT|uΔt|2ϕ(x)dxdt]}<∞. Hence, there exists a Young measure $$\nu = \nu_{t,x,\omega}$$ such that for any Carathéodory function $$\Psi$$ satisfying $$\Psi({u_{{{{\it {\Delta}} t}}}}j,\cdot) \rightharpoonup \overline{\Psi}$$ in $$L^1({\it {\Pi}}_T \times {\it {\Omega}},\phi\,\mathrm{d}x \otimes \mathrm{d}t \otimes {\rm d}P)$$, it follows that   Ψ¯(t,x,ω)=∫RΨ(ξ,t,x,ω)dνt,x,ω(ξ). Define (Panov, 1996; Eymard et al., 2000)   u(t,x,α,ω):=inf{ξ∈R:νt,x,ω((−∞,ξ])>α}. The representation (5.2) follows from the relation $${\mathcal{L}} \circ u^{-1}(t,x,\cdot,\omega) = \nu_{t,x,\omega}$$, where $${\mathcal{L}}$$ denotes the Lebesgue measure on $$[0,1]$$. For predictability and the fact that $$u \in L^p([0,T] \times {\it {\Omega}};L^p({\mathbb{R}}^d \times [0,1],\phi))$$ see Karlsen & Storrøsten (2017, Theorem 4.1); Panov (1996, Section 3); Bauzet et al. (2012, Section A.3.3). 2. Independence of interpolation. Denote by $$v$$ the limit of $$\left\{{{v_{{{{\it {\Delta}} t}}}}}\right\}$$; see step 1. We want to show that $$v = u$$. By Pedregal (1997, Lemma 6.3), this holds true if   T(Δt):=E[∬ΠT|uΔt(t,x)−vΔt(t,x)|ϕ(x)dtdx]→0 as Δt↓0. (5.3) To see this, observe that   T ≤E[∑n=0N−1∫tntn+1∫Rd|uΔt(t,x)−uΔt((tn)+,x)|ϕ(x)dtdx] +E[∑n=0N−1∫tntn+1∫Rd|vΔt((tn+1)−,x)−vΔt(t,x)|ϕ(x)dtdx] =:T1+T2. By Proposition 4.8 (i),   T1≤CT,ϕ∑n=0N−1∫tntn+1t−tndt=23CT,ϕTΔt. By Proposition 4.8 (ii),   T2≤CT,ϕ∑n=0N−1∫tntn+1(tn+1−t)κdt≤CT,ϕTΔtκ, where $$\kappa$$ is defined in Proposition 4.8. This proves (5.3). 3. Entropy inequality. We need to prove that $$u$$ is a Young measure-valued entropy solution in the sense of Karlsen & Storrøsten (2017, Definition 2.2). The result then follows from Karlsen & Storrøsten (2017, Theorem 5.1). Let $$S,V,{\varphi}$$ be as in Lemma 5.1 and define   TΔt:=∑n=0N−1E[∫tntn+1∫Rdbig(S(vΔt(t,x)−V)−S(vΔt((tn+1)−,x)−V))∂tφdxdt]. We want to show that $${\mathscr{T}}_{{{\it {\Delta}} t}} \rightarrow 0$$ as $${{{\it {\Delta}} t}} \downarrow 0$$. Recall the definition of the weighted $$L^\infty$$-norm (2.1). By Proposition 4.8,   |TΔt| ≤‖S‖Lipsupt∈[0,T]{‖∂tφ‖∞,ϕ−1} ×∑n=0N−1E[∫tntn+1∫Rd|vΔt(t,x)−vΔt((tn+1)−,x)|ϕ(x)dxdt] ≤‖S‖Lipsupt∈[0,T]{‖∂tφ‖∞,ϕ−1}CT,ϕTΔtκ, as in the proof of step 2. Concerning the remaining terms in Lemma 5.1, the limit $${{{\it {\Delta}} t}} \downarrow 0$$ is treated exactly as in Karlsen & Storrøsten (2017, Proof of Theorem 4.1, Step 2). It follows that $$u$$ is a Young measure-valued entropy solution. □ 6. Error estimate We now restrict our attention to the case   σ(x,u)=σ(u),σ∈L∞. (𝓐σ,2 As mentioned in Section 1, for homogeneous noise functions $$\sigma = \sigma(u)$$, whenever $${E \left[{\left\Vert{\nabla u_0}\right\Vert_{1,\phi}}\right]} < \infty$$, the entropy solution $$u$$ to (1.1) satisfies a spatial BV estimate of the form   E[∫Rd|∇u(t,x)|ϕ(x)dx]≤C(0≤t≤T) (6.1) for some finite constant $$C$$ (depending on $$u_0,f,\phi,\sigma,T$$). Here $$\nabla u(t,\cdot)$$ is a (locally finite) measure and $$\phi \in \mathfrak{N}$$. This can be seen as a consequence of the fractional space translation estimate (2.3) and Remark 4.7. A direct verification of (6.1) can also be found in Chen et al. (2012, Theorem 2.1) (when $$\phi \equiv 1$$). The same estimate is available for the operator splitting solution; cf. Proposition 4.4. For the error estimate, we consider yet another time interpolation $$\eta_{{{\it {\Delta}} t}}$$ of the operator splitting $$\left\{{u^n}\right\}_{n=0}^N$$. Inspired by Langseth et al. (1996), let   ηΔt(t):=(SSDE(t,tn)−I)SCL(Δt)un⏟uΔt(t)−SCL(Δt)un+SCL(t−tn)un,⏟vΔt(t)t∈[tn,tn+1]. (6.2) A graphical representation of the interpolation $$\eta_{{{\it {\Delta}} t}}$$ is given in Fig. 2. Fig. 2. View largeDownload slide A graphical representation of $$\eta_{{{\it {\Delta}} t}}$$. The value of $$\eta_{{{\it {\Delta}} t}}(t)$$ corresponds to summing (with signs) the values taken at the unfilled dots. Fig. 2. View largeDownload slide A graphical representation of $$\eta_{{{\it {\Delta}} t}}$$. The value of $$\eta_{{{\it {\Delta}} t}}(t)$$ corresponds to summing (with signs) the values taken at the unfilled dots. Theorem 6.1 Fix $$\phi \in \mathfrak{N}$$. Suppose (𝓐f), (𝓐f,1), (𝓐σ) and (𝓐σ,2) are satisfied. Suppose also that $$u^0,u_0 \in L^2({\it {\Omega}},{\mathscr{F}}_0,P;L^2({\mathbb{R}}^d,\phi))$$ satisfy (4.23) with $${{\kappa_0}} = 1$$. Let $$u$$ be the entropy solution of (1.1) and (1.2) according to Definition 2.1 with initial condition $$u_0$$, and let $$\eta_{{{\it {\Delta}} t}}$$ be defined by (6.2). Then there exists a constant $$C$$, independent of $${{{\it {\Delta}} t}}$$ but dependent on $$\sigma,f,T,\phi,u_0,u^0$$, such that   E[‖u(t)−ηΔt(t)‖1,ϕ]≤eCϕ‖f‖Lipt(E[‖u0−u0‖1,ϕ]+CΔt1/3),t∈[0,T]. The proof is split into several parts, the results of which are gathered toward the end of the section. To help motivate the upcoming technical arguments, let us outline a ‘high-level’ overview of the main idea, assuming that all relevant functions are smooth in $$x$$ and the spatial dimension is $$d = 1$$. The function $$\eta_{{{\it {\Delta}} t}}$$ defined in (6.2) ought to satisfy an ‘approximate’ entropy inequality. Formally, we have   dηΔt+∂xf(vΔt)dt=σ(uΔt)dB, (6.3) indicating that the error terms can be expressed as perturbations of the coefficients $$f,\sigma$$. Let $$u$$ be a smooth (in $$x$$) solution of (1.1). By (6.3),   d(ηΔt−u)=−∂x(f(vΔt)−f(u))dt+(σ(uΔt)−σ(u))dB, and thus the Itô formula gives   dS(ηΔt−u) =−S′(ηΔt−u)∂x(f(vΔt)−f(u))dt+S′(ηΔt−u)(σ(uΔt)−σ(u))dB +12S″(ηΔt−u)(σ(uΔt)−σ(u))2dt, for any $$S \in C^2({\mathbb{R}})$$. Upon adding and subtracting identical terms and taking expectations, we arrive at   E[dS(ηΔt−u)]= −E[S′(ηΔt−u)∂x(f(ηΔt)−f(u))dt] +12E[S″(ηΔt−u)(σ(ηΔt)−σ(u))2dt] +E[S′(ηΔt−u)∂x(f(ηΔt)−f(vΔt))dt] +E[S″(ηΔt−u)(∫ηΔtuΔt(σ(z)−σ(u))σ′(z)dz)dt]. The first two terms vanish as $$S\rightarrow \left|{\cdot}\right|$$. Note that these terms also appear in the uniqueness argument, when two exact solutions are compared. Accordingly, they should not be thought of as error terms originating from the splitting procedure. The last two terms, however, are genuine error terms associated with the operator splitting and the interpolation $$\eta_{{{\it {\Delta}} t}}$$. All of the above terms may be recognized in the forthcoming Lemma 6.3. The above simplified representation provides intuition on how to estimate these error terms. This is, in particular, the case concerning the third term on the right-hand side. To this end, note that   ηΔt−vΔt=uΔt−SCL(Δt)un=∫tntσ(uΔt(s))dB(s) for $$t_n \leq t < t_{n+1}$$. Consequently,   ∂x(f(ηΔt)−f(vΔt)) =(f′(ηΔt)−f′(vΔt))∂xvΔt +f′(ηΔt)∫tnt∂xσ(uΔt(s))dB(s). (6.4) Furthermore,   E[|(f′(ηΔt)−f′(vΔt))∂xvΔt|]≤‖f′‖LipE[E[|∫tntσ(uΔt(s))dB(s)||Ftn]|∂xvΔt|], which provides a way to estimate the term since $${v_{{{{\it {\Delta}} t}}}}(t)\in BV$$ and $$\sigma\in L^\infty$$. Because of the lack of regularity, we will work with an approximation of $$\eta_{{{\it {\Delta}} t}}$$. Given $$\left\{{w^n = w^n(x)}\right\}_{n = 0}^{N-1}$$, we set   ψ(t):=(SSDE(t,tn)−I)wn,t∈[tn,tn+1), (6.5) and $${{\tilde{\eta}}} := \psi + {v_{{{{\it {\Delta}} t}}}}$$. Note that $${\eta_{{{\it {\Delta}} t}}} = \psi + {v_{{{{\it {\Delta}} t}}}}$$ whenever $$w^n = {\mathcal{S}_{\text{CL}}}({{{\it {\Delta}} t}})u^n$$ for $$n = 0,\dots,N-1$$. However, because of the lack of differentiability of $${\mathcal{S}_{\text{CL}}}({{{\it {\Delta}} t}})u^n$$, we will work with a sequence $$\left\{{w^n_k}\right\}_{k \geq 1}$$ of smooth functions satisfying $$w^n_k \rightarrow {\mathcal{S}_{\text{CL}}}({{{\it {\Delta}} t}})u^n$$ in $$L^1({\it {\Omega}};L^1({\mathbb{R}}^d,\phi))$$ as $$k \rightarrow \infty$$. To simplify notation, we suppress the dependence on $$k$$ and write $$w^n = w^n_k$$. Proposition 6.2 Suppose (𝓐f), (𝓐σ) and (𝓐σ,2) are satisfied. Let $${{\tilde{\eta}}} = \psi + {v_{{{{\it {\Delta}} t}}}}$$, where $$\psi$$ and $${v_{{{{\it {\Delta}} t}}}}$$ are defined in (6.5) and (3.5), respectively. Then, for all non-negative $$\phi \in C^\infty_c([t_n,t_{n+1}]\times {\mathbb{R}}^d)$$, any $$V \in {\mathbb{D}}^{1,2}$$, and all entropy/entropy–flux pairs $$(S,Q) \in \mathscr{E}$$,   E[∫RdS(η~(tn,x)−V)ϕ(tn,x)dx] −E[∫RdS(η~((tn+1)−,x)−V)ϕ(tn+1,x)dx] +E[∬ΠnS(η~−V)∂tϕ+Q(η~,V)⋅∇ϕdxdt] +E[∬Πn∫Vη~S′(z−V)(f′(z−ψ)−f′(z))dz⋅∇ϕdxdt] +E[∬Πn∫Vη~S″(z−V)f′(z−ψ)dz⋅∇ψϕdxdt] −E[∬ΠnS″(η~−V)DtVσ(ψ+wn)ϕdxdt] +12E[∬ΠnS″(η~−V)σ2(ψ+wn)ϕdxdt]≥0, where $${\it {\Pi}}_n = [t_n,t_{n+1}] \times {\mathbb{R}}$$. The proof of Proposition 6.2 is deferred to Section A.1. To ensure that the relevant quantities are Malliavin differentiable, we replace the entropy solution $$u$$ by the viscous approximation $${{u^\varepsilon}}$$, which solves   duε+∇⋅f(uε)dt=σ(x,uε)dB(t)+εΔuεdt,uε(0)=u0, and then send $$\varepsilon \downarrow 0$$ at a later stage. Let us recall that $$\left\{{D_r{{u^\varepsilon}}(t)}\right\}_{t > r}$$ is a predictable weak solution to the linear problem   dw+∇⋅(f′(uε)w)dt=σ′(x,uε)wdB(t)+εΔwdt,w(r)=σ(uε(r)), for almost all $$r \in [0,T]$$ (cf. Karlsen & StorrØsten, 2017, Section 3). Furthermore,   esssupr∈[0,T]supt∈[0,T]{E [ ∥Druε(t)∥2,ϕ2}<∞. As a consequence of Karlsen & StorrØsten (2017, Theorem 5.1) and Pedregal (1997, Proposition 6.12), we have $${{u^\varepsilon}} \rightarrow u$$ in $$L^1([0,T] \times {\it {\Omega}};L^1({\mathbb{R}}^d,\phi))$$ as $$\varepsilon \downarrow 0$$. In fact, under the assumptions of Theorem 6.1, $${{u^\varepsilon}} \rightarrow u$$ with rate $$1/2$$ (Chen et al., 2012, Theorem 5.2). We may now proceed with the doubling-of-the-variables argument. Lemma 6.3 Fix $$\phi \in \mathfrak{N}$$. Let $${{u^\varepsilon}} = {{u^\varepsilon}}(s,y)$$ be the viscous approximation of (1.1). Take $$w(t,x) = w^n(x)$$ for $$t \in [t_n,t_{n+1})$$, and let $$\psi = \psi(t,x)$$, $${v_{{{{\it {\Delta}} t}}}} = {v_{{{{\it {\Delta}} t}}}}(t,x)$$ and $${{\tilde{\eta}}} = {{\tilde{\eta}}}(t,x)$$ be defined in Proposition 6.2. Let $$t_0 \in [0,T)$$, and pick $${\it {\gamma}},r_0,r > 0$$ such that $$t_0 \leq T-2({\it {\gamma}} + r_0)$$. Define   ξγ(t)=1−∫0tJγ+(s−t0)ds. Furthermore, let   φ(t,x,s,y)=12dϕ(x+y2)Jr(x−y2)Jr0+(s−t)ξγ(t), and $$S_\delta$$ be defined in (4.9). Then   L−R+F+T1+T2+T3+T4+T5+T6≥0, (6.6) where   L =E[∬ΠT∫RdSδ(η~(0,x)−uε(s,y))φ(0,x,s,y)dxdsdy],R =−E[⨌ΠT2Sδ(η~−uε)(∂t+∂s)φdX],F =E[⨌ΠT2Q(uε,η~)⋅∇yφ+Q(η~,uε)⋅∇xφdX],T1 =12E[⨌ΠT2Sδ″(uε−η~)(σ(uε)−σ(η~))2φdX],T2 =E[⨌ΠT2Sδ″(uε−η~)(σ(uε)−Dtuε)σ(ψ+w)φdX],T3 =E[⨌ΠT2Sδ″(uε−η~)(∫η~ψ+w(σ(z)−σ(uε))σ′(z)dz)φdX],T4 =E[⨌ΠT2∫uεη~Sδ′(z−uε)(f′(z−ψ)−f′(z))dz⋅∇xφdX] +E[⨌ΠT2∫uεη~Sδ″(z−uε)f′(z−ψ)dz⋅∇xψφdX],T5 =εE[⨌ΠT2Sδ(uε−η~)ΔyφdX],T6 =∑n=0N−1E[∬ΠT∫Rd(Sδ(η~((tn+1),x)−uε(s,y)) −Sδ(η~((tn+1)−,x)−uε(s,y)))φ(tn+1,x,s,y)dxdsdy], where $$\mathrm{d}X = \mathrm{d}x\,\mathrm{d}t\,\mathrm{d}s\,\mathrm{d}y$$. Proof. Let us first assume $$\phi \in C^\infty_c({\mathbb{R}}^d)$$, as the result for $$\phi\in \mathfrak{N}$$ then follows from an approximation argument. After a standard application of Itô’s formula to $${{u^\varepsilon}}(s,y) \mapsto S_\delta({{u^\varepsilon}}(s,y)-{{\tilde{\eta}}}(t,x)) \varphi(s)$$ for $$s \geq t$$, we arrive at   E[⨌ΠT2Sδ(uε−η~)∂sφ+Q(uε,η~)⋅∇yφdX] +12E[⨌ΠT2Sδ″(uε−η~)σ2(uε)φdX]+εE[⨌ΠT2Sδ(uε−η~)ΔyφdX]≥0; cf. Karlsen & Storrøsten (2017, Lemma 5.3). Take $$V = {{u^\varepsilon}}(s,y)$$ in Proposition 6.2, integrate in $$(s,y) \in {\it {\Pi}}_T$$ and sum over $$n = 0,\dots,N-1$$. The outcome is   E[∬ΠT∫RdSδ(η~(0,x)−uε(s,y))φ(0,x,s,y)dxdsdy] +E[⨌ΠT2Sδ(η~−uε)∂tφ+Q(η~,uε)⋅∇xφdX] +E[⨌ΠT2∫uεη~Sδ′(z−uε)(f′(z−ψ)−f′(z))dz⋅∇xφdX] +E[⨌ΠT2∫uεη~Sδ″(z−uε)f′(z−ψ)dz⋅∇xψφdX] −E[⨌ΠT2Sδ″(η~−uε)Dtuεσ(ψ+w)φdX] +12E[⨌ΠT2Sδ″(η~−uε)σ2(ψ+w)φdX] +∑n=0N−1E[∬ΠT∫Rd(Sδ(η~((tn+1),x)−uε(s,y)) −Sδ(η~((tn+1)−,x)−uε(s,y)))φ(tn+1,x,s,y)dxdsdy]≥0. The lemma follows upon adding the two previous inequalities, noting that   12σ2(uε)−Dtuεσ(ψ+w)+12σ2(ψ+w) =12(σ(ψ+w)−σ(uε))2+(σ(uε)−Dtuε)σ(ψ+w) =12(σ(η~)−σ(uε))2+∫η~ψ+w(σ(z)−σ(uε))σ′(z)dz +(σ(uε)−Dtuε)σ(ψ+w). □ In the following we estimate the terms appearing in Lemma 6.3. The underlying assumptions are the ones made in Theorem 6.1. We let $$C$$ denote a generic constant, meaning that it is independent of the ‘small’ parameters $${{{\it {\Delta}} t}},r,r_0,{\it {\gamma}},\varepsilon,\delta$$. Furthermore, given a term $${\mathscr{T}}$$, we write $${\mathscr{T}} = \mathcal{O}(g({{{\it {\Delta}} t}}, \dots,\delta))$$ whenever $$\left|{{\mathscr{T}}}\right| \leq Cg({{{\it {\Delta}} t}}, \dots,\delta)$$ for some non-negative function $$g$$. Estimate 6.4 Let $$L$$ be defined in Lemma 6.3. Then   lim supr0↓0L≤E[‖u0−u0‖1,ϕ]+O(δ+r). Proof. By (4.10),   |Sδ(η~(0,x)−uε(s,y))−|η~(0,x)−uε(s,y)||≤δ. By the reverse triangle inequality,   ||η~(0,x)−uε(s,y)|−|η~(0,x)−u0(y)|| ≤|uε(s,y)−u0(y)|,||η~(0,x)−u0(y)|−|η~(0,x)−u0(x)|| ≤|u0(y)−u0(x)|. Hence, after adding and subtracting identical terms, noting that $${{\tilde{\eta}}}(0) = u^0$$, it follows by the triangle inequality that   |Sδ(η~(0,x)−uε(s,y))−|u0(x)−u0(x)||≤δ+|uε(s,y)−u0(y)|+|u0(y)−u0(x)|. By (4.15),   |L−E[‖u0−u0‖1,ϕ⋆Jr]| ≤δ‖ϕ‖L1(Rd)+∫0TE[‖uε(s)−u0‖1,ϕ⋆Jr]Jr0+(s)ds⏟Z1 +E[12d∬Rd×Rd|u0(y)−u0(x)|ϕ(x+y2)Jr(x−y2)dxdy]Z2⏟. Thanks to Karlsen & Storrøsten (2017, Lemma 2.3), $${\mathscr{Z}}_1 \to 0$$ as $$r_0 \to 0$$. Regarding $${\mathscr{Z}}_2$$ we apply (4.14). As $$u_0$$ satisfies (4.23) with $${{\kappa_0}} = 1$$,   Z2=E[∬Rd×Rd|u0(x+z)−u0(x−z)|ϕ(x)Jr(z)dxdz]=O(r). Finally, we apply Lemma A.8 to conclude that   |E[‖u0−u0‖1,ϕ⋆Jr−‖u0−u0‖1,ϕ]|=O(r). □ Estimate 6.5 Let $$R$$ be defined in Lemma 6.3. Then   lim infε,r0↓0R≥E[∫0T‖η~(t)−u(t)‖1,ϕJγ+(t−t0)dt]+O(δ+r). Proof. It is easy to check that   R=E[⨌ΠT2Sδ(η~(t,x)−uε(s,y))12dϕ(x+y2)×Jr(x−y2)Jr0+(s−t)Jγ+(t−t0)dX]. Moreover, adding and subtracting identical terms, we obtain   |Sδ(η~(t,x)−uε(s,y))−|η~(t,x)−uε(t,x)||≤δ+|uε(s,y)−uε(t,y)|+|uε(t,y)−uε(t,x)|, and so   |R−E[∫0T‖η~(t)−uε(t)‖1,ϕ⋆JrJγ+(t−t0)dt]| ≤δ‖ϕ‖L1(Rd)+E[∬[0,T]2‖uε(s)−uε(t)‖ϕ⋆JrJr0+(s−t)Jγ+(t−t0)dsdt]⏟Z1 +E[∬ΠT∫Rd|uε(t,y)−uε(t,x)|12dϕ(x+y2)Jr(x−y2)Jγ+(t−t0)dxdydt]⏟Z2. Because of Lemma A.9, $$\lim_{r_0 \downarrow 0} {\mathscr{Z}}_1 = 0$$. Next, we utilize the strong convergence $${{u^\varepsilon}} \rightarrow u$$ in $$L^1([0,T] \times {\it {\Omega}};L^1({\mathbb{R}}^d,\phi))$$ and (4.14) to conclude that   limε,r0↓0Z2=∫0TE[∬Rd×Rd|u(t,x+z)−u(t,x−z)|ϕ(x)Jr(z)dxdz]Jγ+(t−t0)dt. It follows from Karlsen & Storrøsten (2017, Proposition 5.2) and the assumption (4.23) with $${{\kappa_0}} = 1$$ that $$\left|{\lim_{\varepsilon,r_0 \downarrow 0} {\mathscr{Z}}_2}\right| = \mathcal{O}(r)$$. The claim is now a consequence of Lemma A.8. □ Estimate 6.6 Let $$F$$ be defined in Lemma 6.3. Then   lim supε,r0↓0F≤Cϕ‖f‖LipE[∫0T‖u(t)−η~(t)‖1,ϕξγ(t)dt]+O(δ(1+1r)+r). Proof. Observe that   F=F1+F2+F3, (6.7) where   F1 :=E[⨌ΠT2Sδ′(uε−η~)(f(uε)−f(η~))(∇x+∇y)φdX],F2 :=−E[⨌ΠT2∫uεη~Sδ″(z−uε)(f(z)−f(uε))dz⋅∇xφdX],F3 :=−E[⨌ΠT2∫η~uεSδ″(z−η~)(f(z)−f(η~))dz⋅∇yφdX]. The decomposition (6.7) follows from the identities   Qδ(uε,η~) =Sδ′(uε−η~)(f(uε)−f(η~))−∫η~uεSδ″(z−η~)(f(z)−f(η~))dz,Qδ(η~,uε) =Sδ′(η~−uε)(f(η~)−f(uε))−∫uεη~Sδ″(z−uε)(f(z)−f(uε))dz, derived using integration by parts. Next, we claim that   |F2|+|F3|=O(δ(1+1r)). (6.8) We consider $$F_2$$; the $$F_3$$ term is estimated likewise. Note that   |∫uεη~Sδ″(z−uε)(f(z)−f(uε))dz|≤‖f‖Lipδ. Hence,   |F2|≤‖f‖LipδE[⨌ΠT2|∇xφ|dX]. By a straightforward computation,   ⨌ΠT2|∇xφ|dX≤12T(Cϕ+‖∇J‖L1(Rd)1r)‖ϕ‖L1(Rd). This proves (6.8). Next, we claim that   lim supε,r0↓0F1≤Cϕ‖f‖LipE[∫0T‖u(t)−η~(t)‖1,ϕ∗Jrξγ(t)dt]+O(δ+r). (6.9) Set   Fδ(b,a)=Sδ′(b−a)(f(b)−f(a)). Then   |Fδ(b,a)−Fδ(c,a)| =|∫cb∂z(Sδ′(z−a)(f(z)−f(a)))dz| ≤2‖f‖Lipδ+‖f‖Lip|b−c|; whence   |Fδ(uε(s,y),η~(t,x))−Fδ(uε(t,x),η~(t,x))|≤‖f‖Lip(2δ+|uε(s,y)−uε(t,y)|+|uε(t,y)−uε(t,x)|), and so   F1−E∬ΠTFδ(uε(t,x),η~(t,x))⋅(∇ϕ∗Jr)(x)ξγ(t)dxdt ≤Cϕ‖f‖LipE[∬[0,T]2‖uε(s)−uε(t)‖1,ϕ⋆JrJr0+(s−t)ξγ(t)dsdt]  +Cϕ‖f‖LipE[∫0T∬Rd×Rd|uε(t,x+z)−uε(t,x−z)|Jr(z)ξγ(t)ϕ(x)dxdzdt]  +2δ‖f‖LipT‖∇ϕ‖L1(Rd), where we have made a change of variables as in Estimate 6.5. Following the same reasoning as in that estimate we arrive at   lim supε,r0↓0F1≤E[∬ΠTFδ(u(t,x),η~(t,x))⋅(∇ϕ∗Jr)(x)ξγ(t)dxdt]+O(δ+r). Inequality (6.9) follows from $$\mathcal{F}_\delta(a,b) \le \left\Vert{f}\right\Vert_\mathrm{Lip}\left|{a-b}\right|$$ and $$\left|{\nabla \phi}\right| \le C_\phi\phi$$. Combining the above estimates for $$F_1,F_2,F_3$$ concludes the proof of the claim. □ Estimate 6.7 Let $${\mathscr{T}}_1$$ be defined in Lemma 6.3. Then   |T1|≤Cδ. Proof. Since $$S_\delta'' = 2J_\delta$$,   Sδ″(uε−η~)(σ(uε)−σ(η~))2≤2‖σ‖Lip2Jδ(uε−η~)|uε−η~|2≤2‖σ‖Lip2‖J‖∞δ. Because of (4.15) and Young’s inequality for convolutions,   ⨌ΠT2φdX=(∫0T∫0TJr0+(s−t)ξγ(t)dsdt)(∫Rdϕ⋆Jr(x)dx)≤T‖ϕ‖L1(Rd). The result follows. □ Estimate 6.8 Let $${\mathscr{T}}_2$$ be defined in Lemma 6.3. Then   limr0↓0T2=0. Proof. This follows exactly as in Karlsen & StorrØsten (2017, Limit 5). However, the assumption $$\sigma \in L^\infty$$ simplifies the analysis and allows for $$\phi \in \mathfrak{N}$$ instead of $$C^\infty_c({\mathbb{R}}^d)$$. □ Estimate 6.9 Let $${\mathscr{T}}_3$$ be defined in Lemma 6.3. Then   |T3|≤C1δE[∑n=0N−1∫tntn+1‖wn−vΔt(t)‖ϕ⋆Jrdt]. Proof. Now, as $${{\tilde{\eta}}} = \psi + {v_{{{{\it {\Delta}} t}}}}$$,   |∫η~ψ+w(σ(z)−σ(uε))σ′(z)dz|≤2‖σ‖∞‖σ‖Lip|w−vΔt|. Keep in mind that $$w(t) = w^n$$ for $$t \in [t_n,t_{n+1})$$. The estimate then follows from (4.10) and (4.15). □ Estimate 6.10 Let $${\mathscr{T}}_4$$ be defined in Lemma 6.3. Then   |T4|≤CΔt(1+E[∫0T‖∇w(t)‖1,ϕ⋆Jr]). Proof. The estimate is established under the assumption that $${v_{{{{\it {\Delta}} t}}}}$$ is smooth in $$x$$. The general result follows by an approximation argument. Integrating by parts and using the chain rule,   T4 =E[⨌ΠT2∫uεη~Sδ′(z−uε)(f′(z−ψ)−f′(z))dz⋅∇xφdX] =−E[⨌ΠT2Sδ′(η~−uε)(f′(vΔt)−f′(η~))⋅∇xη~φdX] +E[⨌ΠT2∫uεη~Sδ′(z−uε)f″(z−ψ)dz⋅∇xψφdX]. Next, we observe that   ∫uεη~Sδ′(z−uε)f″(z−ψ)dz=−∫uεη~Sδ″(z−uε)f′(z−ψ)dz+Sδ′(η~−uε)f′(vΔt). Therefore,   T4=E[∫∫∫∫ΠT2⁡Sδ′(η~−uε)f′(η~)⋅∇xψφdX]⏟Z1  +E∫∫∫∫ΠT2⁡Sδ′(η~−uε)(f′(η~)−f′(vΔt))⋅∇xvΔtφdX⏟Z2; cf. (6.4). Consider $${\mathscr{Z}}_2$$. Since $${v_{{{{\it {\Delta}} t}}}}(t)$$ is $${\mathscr{F}}_{t_n}$$-measurable for all $$t \in [t_{n},t_{n+1})$$,   |Z2|≤E[∫∫∫∫ΠT2⁡|f′(η~)−f′(vΔt)||∇xvΔt|φdX] ≤‖f′‖Lip∑n=0N−1∫∫∫∫ΠT×Πn[EE[|ψ||Ftnn]|∇xvΔt|]φdX. By definition,   ψ(t,x)=∫tntσ(ψ(r,x)+wn(x))dB(r),tn≤t<tn+1. (6.10) In view of Jensen’s inequality for conditional expectation and the conditional Itô isometry (Capiński et al., 2012, Theorem 3.20),   E[|ψ(t,x)||Ftn] ≤E[|ψ(t,x)|2|Ftn]1/2 =E[∫tntσ2(ψ(t,x)+wn(x))ds|Ftn]1/2 ≤‖σ‖∞t−tn. It follows from Proposition 4.4 that   |Z2|≤‖σ‖∞‖f′‖LipΔtE [∫0T‖∇xvΔt(t)‖1,ϕ⋆Jrdt]≤CΔt. Consider $${\mathscr{Z}}_1$$. In view of (4.15),   |Z1|≤‖f‖LipE[⨌ΠT2|∇xψ|φdX]≤‖f‖LipE[∬ΠT|∇xψ|(ϕ⋆Jr)dxdt]. Differentiating (6.10) yields, for $$t_n \le t < t_{n+1}$$,   ∇xψ(t,x)=∫tntσ′(ψ(r,x)+wn(x))(∇xψ(r,x)+∇xwn(x))dB(r). By Lemma 6.11 below there is a constant $$C > 0$$, depending only on $$\sigma$$, such that   E[|∇xψ(t,x)|]≤Ct−tnE[|∇wn(x)|],tn≤t<tn+1. We conclude that   |Z1|≤C(E[∫0T‖∇w(t)‖1,ϕ∗Jr]dt)Δt. □ Lemma 6.11 Suppose $$h:[t_n,t_{n+1}] \times {\it {\Omega}} \rightarrow {\mathbb{R}}^d$$ is predictable and   P[∫tnt|h(s)|2ds<∞]=1. Suppose $$X(t_n) \in L^p({\it {\Omega}},{\mathscr{F}}_{t_n},P;{\mathbb{R}}^d)$$, $$1 \le p < \infty$$ and let $$X:[t_n,t_{n+1}] \times {\it {\Omega}} \rightarrow {\mathbb{R}}^d$$ satisfy   X(t)=X(tn)+∫tnth(s)dB(s),t∈[tn,tn+1]. Suppose there exist a constant $$K$$ and $$Y \in L^p({\it {\Omega}},{\mathscr{F}}_{t_n},P)$$ such that   |h(t;ω)|≤Y(ω)+K|X(t)|,t∈[tn,tn+1]. (6.11) Then, for all $$t \in [t_n,t_{n+1}]$$ and $$\beta > p(c_p^{1/p}K)^2/2$$,   suptn≤s≤tE[|X(s)|p]1/p≤C(β)eβ(t−tn)(E[|X(tn)|p]1/p+cp1/pt−tnE[|Y|p]1/p), where $$C(\beta) = \left(1-c_p^{1/p}K\sqrt{p/2\beta}\right)^{-1}$$ and $$c_p$$ is the constant from the Burkholder–Davis–Gundy inequality. Proof. Set   ‖X‖β,p,τ:=(suptn≤t≤τe−β(t−tn)E[|X(t)|p])1/p. The triangle inequality yields   E[|X(t)|p]1/p≤E[|∫tnth(s)dB(s)]|p1/p+E[|X(tn)|p]1/p. By the Burkholder–Davies–Gundy inequality,   E[|∫tnth(s)dB(s)|p]1/p≤cp1/pE[(∫tnth2(s)ds)p/2]1/p. Because of (6.11) and the triangle inequality on $$L^p({\it {\Omega}};L^2([t_n,t]))$$,   E[(∫tnt|h(s)|2ds)p/2]1/p≤t−tnE[|Y|p]1/p+KE[(∫tnt|X(s)|2ds)p/2]1/p. By Minkowski’s integral inequality,   E[(∫tnt|X(s)|2ds)p/2]2/p≤∫tntE[|X(s)|p]2/pds. Furthermore,   ∫tntE[|X(s)|p]2/pds =e2β(t−tn)/p∫tnt(e−β(t−s)e−β(s−tn)E[|X(s)|p])2/pds ≤e2β(t−tn)/p‖X‖β,p,t2∫tnte−2β(t−s)/pds =p2β(e2β(t−tn)/p−1)‖X‖β,p,t2. Summarizing, we arrive at   E[|X(t)|p]1/p ≤E[|X(tn)|p]1/p+cp1/pt−tnE[|Y|p]1/p +cp1/pKp2β(e2β(t−tn)/p−1)1/2‖X‖β,p,t. Multiplying by $$\,{\it e}^{-\beta(t-t_n)/p}$$ and taking the supremum over $$t_n \leq t \leq \tau$$, we obtain   ‖X‖β,p,τ≤E[|X(tn)|p]1/p+cp1/pτ−tnE[|Y|p]1/p+cp1/pKp2β‖X‖β,p,τ. Choosing $$\beta$$ sufficiently large, i.e., $$c_p^{1/p}K\sqrt{p/2\beta} < 1$$, we secure the bound   ‖X‖β,p,τ≤11−cp1/pKp/2β(cp1/pτ−tnE[|Y|p]1/p+E[|X(tn)|p]1/p). The result follows upon multiplication by $$e^{\beta(\tau-t_n)/p}$$, since   eβ(τ−tn)/p‖X‖β,p,τ=(suptn≤t≤τeβ(τ−t)E[|X(t)|p])1/p≥suptn≤t≤τE[|X(t)|p]1/p. □ Estimate 6.12 Let $${\mathscr{T}}_5$$ be defined in Lemma 6.3. Then   T5=O(ε). Proof. This follows as in Karlsen & Storrøsten (2017, Limit 6). □ Estimate 6.13 Let $${\mathscr{T}}_6$$ be defined in Lemma 6.3. Then   |T6|≤2∑n=0N−1E[‖SCL(Δt)un−wn‖1,ϕ⋆Jr]. Proof. First, we note that $$\left|{S_\delta(b)-S_\delta(a)}\right| \le \left|{b-a}\right|$$. This and (4.15) yields   |T6|≤∑n=0N−1E[‖η~(tn+1)−η~((tn+1)−)‖1,ϕ⋆Jr]. Since   η~(tn+1)−η~((tn+1)−)=SSDE(tn+1,tn)(SCL(Δt)un−wn)+SCL(Δt)un−wn, the result follows from (3.2). □ Proof of Theorem 6.1. Consider Lemma 6.3 and take the upper limits in (6.6) as $$r_0 \downarrow 0, \varepsilon \downarrow 0$$ and $${\it {\gamma}} \downarrow 0$$ (in that order). Next we recall that $$w^n = w^n_k$$. Letting $$k \rightarrow \infty$$, then $$w^n_k \rightarrow {\mathcal{S}_{\text{CL}}}({{{\it {\Delta}} t}})u^n$$ in $$L^1({\it {\Omega}},L^1({\mathbb{R}}^d,\phi))$$. Because of the $$L^1$$-Lipschitz continuity of $${\mathcal{S}_{\text{CL}}}$$ (cf. Proposition 4.8) and the uniform BV-bound on the splitting approximation, it follows from Estimates 6.4–6.10 and 6.12–6.13 that   E[‖u0−u0‖1,ϕ]+Cϕ‖f‖Lip∫0t0E[‖ηΔt(t)−u(t)‖1,ϕ]dt +O(δ+r+Δt+δr+Δtδ)≥E[‖ηΔt(t0)−u(t0)‖1,ϕ]. Finally, we apply Grönwall’s inequality and then choose $$\delta = {{{\it {\Delta}} t}}^{2/3}$$ and $$r = {{{\it {\Delta}} t}}^{1/3}$$. □ A. Appendix A.1 Proof of Proposition 6.2 The proof of Proposition 6.2 is based on the following result. Lemma A.1 Suppose $$u,w \in L^2({\it {\Omega}},P,{\mathscr{F}}_{t_n};L^2({\mathbb{R}}^d))$$ and $$w$$ is smooth. Set   ψ(t)=(SSDE(t,tn)−I)w,v(t)=SCL(t−tn)u,t∈[tn,tn+1]. Then for all $$(S,Q) \in \mathscr{E}$$, all non-negative $${\varphi} \in C^\infty_c({\it {\Pi}}_n^2)$$ and all $$V \in \mathcal{S}$$,   R−L+T1+T2−T3+T4≥0, where   L =E[∬Πn∫RdS(v(tn+1,x)+ψ(s,y)−V)φ(tn+1,x,s,y)dxdyds] +E[∬Πn∫RdS(v(t,x)+ψ(tn+1,y)−V)φ(t,x,tn+1,y)dydxdt],R =E[∬Πn∫RdS(v(tn,x)+ψ(s,y)−V)φ(tn,x,s,y)dxdyds] +E[∬Πn∫RdS(v(t,x)+ψ(tn,y)−V)φ(t,x,tn,y)dydxdt],T1 =E[⨌Πn2S(v(t,x)+ψ(s,y)−V)(∂t+∂s)φdX],T2 =E[⨌Πn2Q(v(t,x),V−ψ(s,y))⋅∇xφdX],T3 =E[⨌Πn2S″(v(t,x)+ψ(s,y)−V)DsVσ(ψ(s,y)+w(y))φdX],T4 =12E[⨌Πn2S″(v(t,x)+ψ(s,y)−V)σ2(ψ(s,y)+w(y))φdX] and $${\it {\Pi}}_n = [t_n,t_{n+1}] \times {\mathbb{R}}^d$$. Proof. The entropy inequality reads    ∫RdS(v(tn,x)−c)φ(tn,x,s,y)−S(v(tn+1,x)−c)φ(tn+1,x,s,y)dx +∬ΠnS(v−c)∂tφ+Q(v,c)⋅∇xφdtdx≥0 (A.1) for all $$c \in {\mathbb{R}}$$ and all $$s,y \in {\it {\Pi}}_n$$. Specify $$c = V-\psi(s,y)$$ in (A.1), integrate in $$(s,y)$$ and take expectations to obtain   E[∬Πn∫RdS(v(tn,x)+ψ(s,y)−V)φ(tn,x,s,y)dxdsdy] −E[∬Πn∫RdS(v(tn+1,x)+ψ(s,y)−V)φ(tn+1,x,s,y)dxdsdy] +E[⨌Πn2S(v+ψ−V)∂tφ+Q(v,V−ψ)⋅∇xφdX]≥0. (A.2) Note that $$v(t)$$ is $${\mathscr{F}}_{t_n}$$-adapted for all $$t \in [t_n,t_{n+1}]$$. To reveal the equation satisfied by $$\psi$$, let $$\zeta(t) = {\mathcal{S}_{\text{SDE}}}(t,t_n)w$$. By definition,   ζ(t,x)=w(x)+∫tntσ(ζ(r,x))dB(r). Since $$\psi(t) = \zeta(t) - w$$,   ψ(t,x)=∫tntσ(ψ(r,x)+w(x))dB(r),t∈[tn,tn+1]. (A.3) Fix $$t,x \in {\it {\Pi}}_n, y \in {\mathbb{R}}^d$$ and set   X(s):=v(t,x)+ψ(s,y),F(X(s),V,s):=S(X(s)−V)φ(t,x,s,y),s∈[tn,tn+1]. By (A.3),   X(s)=v(t,x)+∫tnsσ(ψ(r,y)+w(y))dB(r). By Theorem A.10,   S(X(tn+1)−V)φ(t,x,tn+1,y) =S(X(tn)−V)φ(t,x,tn,y) +∫tntn+1S(X(s)−V)∂sφds +∫tntn+1S′(X(s)−V)σ(ψ(s)+w)φdB(s) −∫tntn+1S″(X(s)−V)DsVσ(ψ(s)+w)φds +12∫tntn+1S″(X(s)−V)σ2(ψ(s)+w)φds, where the stochastic integral is interpreted as a Skorohod integral. Upon integrating in $$t,x,y$$ and taking expectations,   E[∬Πn∫RdS(v(t,x)+ψ(tn,y)−V)φ(t,x,tn,y)dydtdx] −E[∬Πn∫RdS(v(t,x)+ψ(tn+1,y)−V)φ(t,x,tn+1,y)dydtdx] +E[⨌Πn2S(v(t,x)+ψ(s,y)−V)∂sφdX] +12E[⨌Πn2S″(v(t,x)+ψ(s,y)−V)(σ(ψ(s,y)+w(y)))2φdX] −E[⨌Πn2S″(v(t,x)+ψ(s,y)−V)DsVσ(ψ(s,y)+w(y))φdX]=0. (A.4) Adding (A.2) and (A.4) concludes the proof. □ Proof of Proposition 6.2. We use   φ(t,x,s,y)=12dϕ(t+s2,x+y2)Jr(x−y2)Jr0(t−s) (A.5) in Lemma A.1 and then send $$r_0, r$$ to zero (in that order). The sought-after result for $$V \in \mathcal{S}$$ is a consequence of Limits A.2–A.6 below. The extension to $$V \in {\mathbb{D}}^{1,2}$$ follows by an approximation argument as in Karlsen & StorrØsten (2017, Lemma 2.2). □ Limit A.2 Let $$L, R$$ be defined in Lemma A.1 and $${\varphi}$$ in (A.5). Then   limr,r0↓0L(r,r0) =E[∫RdS(v(tn+1,x)+ψ(tn+1,x))ϕ(tn+1,x)dx],limr,r0↓0R(r,r0) =E[∫RdS(v(tn,x)+ψ(tn,x)−V)ϕ(tn,x)dx]. Proof. Let us only consider the term   E[∬Πn∫RdS(v(tn+1,x)+ψ(s,y)−V)φ(tn+1,x,s,y)dxdyds]=:Z. The remaining terms can be treated in the same way. As a consequence of the dominated convergence theorem and Lemma A.9,    limr0↓0Z=12E[∫Rd∫RdS(v(tn+1,x)+ψ(tn+1,y)−V) ×12dϕ(tn+1,x+y2)Jr(x−y2)dxdy]. Moreover,   limr,r0↓0Z=12E[∫RdS(v(tn+1,x)+ψ(tn+1,x)−V)ϕ(tn+1,x)]. □ Limit A.3 Let $${\mathscr{T}}_1$$ be defined in Lemma A.1 and $${\varphi}$$ in (A.5). Then   limr,r0↓0T1=E[∬ΠnS(u(t,x)−V)∂tϕ(t,x)dxdt]. Proof. Observe that   (∂t+∂s)φ(t,x,s,y)=12d∂1ϕ(t+s2,x+y2)Jr(x−y2)Jr0(t−s). The result follows by the dominated convergence theorem and Lemma A.9; consult the proof of Limit A.2. □ Limit A.4 Let $${\mathscr{T}}_2$$ be defined in Lemma A.1 and $${\varphi}$$ in (A.5). Then   limr,r0↓0T2 =E[∬ΠnQ(v+ψ,V)⋅∇ϕdxdt] +E[∬Πn(∫Vv+ψS′(z−V)(f′(z−ψ)−f′(z))dz)⋅∇ϕdxdt] +E[∬Πn(∫Vv+ψS″(z−V)f′(z−ψ)dz)⋅∇ψϕdxdt]. Proof. First observe that   (∇x+∇y)φ(t,x,s,y)=12d∂2ϕ(t+s2,x+y2)Jr(x−y2)Jr0(t−s). Integration by parts results in   T2 =E[⨌Πn2Q(v(t,x),V−ψ(s,y)) ⋅12d∂2ϕ(t+s2,x+y2)Jr(x−y2)Jr0(t−s)dxdtdyds] +E[⨌Πn2∇y⋅Q(v(t,x),V−ψ(s,y))φ(t,x,s,y)dxdtdyds] =:T21+T22. It is straightforward to show that   limr,r0↓0T21 =E[∬ΠnQ(v(t,x),V−ψ(t,x))⋅∇ϕ(t,x)dxdt]. Finally, we apply the identity   Q(v,V−ψ)=Q(v+ψ,V)+∫Vv+ψS′(z−V)(f′(z−ψ)−f′(z))dz. Consider $${\mathscr{T}}_2^2$$. By the chain rule,   T22=−E[⨌Πn2∂2Q(v(t,x),V−ψ(s,y))⋅∇yψ(s,y)φ(t,x,s,y)dxdtdyds]. Sending $$r_0, r$$ to zero, we arrive at   limr,r0↓0T22=−E[∬Πn∂2Q(v(t,x),V−ψ(t,x))⋅∇xψ(t,x)ϕ(t,x)dxdt]. Finally, note that   ∂2Q(v,V−ψ) =−∫V−ψvS″(z−V+ψ)f′(z)dz =−∫Vv+ψS″(z−V)f′(z−ψ)dz. This concludes the proof. □ Limit A.5 Let $${\mathscr{T}}_3$$ be defined in Lemma A.1 and $${\varphi}$$ in (A.5). Then   limr,r0↓0T3=E[∬ΠnS″(v(t,x)+ψ(t,x)−V)DtVσ(ψ(t,x)+w(x))ϕ(t,x)dxdt]. Proof. The proof is a straightforward application of the dominated convergence theorem and Lemma A.9. □ Limit A.6 Let $${\mathscr{T}}_4$$ be defined in Lemma A.1 and $${\varphi}$$ by (A.5). Then   limr,r0↓0T4=12E[∬ΠnS″(v(t,x)+ψ(t,x)−V)σ2(ψ(t,x)+w(x))ϕ(t,x)dxdt]. Proof. This term may be treated similarly to $${\mathscr{T}}_3$$. □ A.2 Weighted $$L^p$$-spaces In the next two lemmas, we collect a few elementary properties of (weight) functions in $$\mathfrak{N}$$. For proofs, see Karlsen & Storrøsten (2017). Lemma A.7 Suppose $$\phi \in \mathfrak{N}$$ and $$0 < p < \infty$$. Then, for $$x,z\in {\mathbb{R}}^d$$,   |ϕ1/p(x+z)−ϕ1/p(x)|≤wp,ϕ(|z|)ϕ1/p(x), where   wp,ϕ(r)=Cϕpr(1+CϕpreCϕr/p), which is defined for all $$r \geq 0$$. As a consequence it follows that if $$\phi(x_0) = 0$$ for some $$x_0 \in {\mathbb{R}}^d$$ then $$\phi \equiv 0$$ (and by definition $$\phi \notin \mathfrak{N}$$). Lemma A.8 Fix $$\phi \in \mathfrak{N}$$, and let $$w_{p,\phi}$$ be defined in Lemma A.7. Let $$J$$ be a mollifier as defined in Section 2 and take $$\phi_\delta = \phi \star J_\delta$$ for $$\delta > 0$$. Then (i) $$\phi_\delta \in \mathfrak{N}$$ with $$C_{\phi_\delta} = C_\phi$$; (ii) for any $$u \in L^p({\mathbb{R}}^d,\phi)$$,   |‖u‖p,ϕp−‖u‖p,ϕδp|≤w1,ϕ(δ)min{‖u‖p,ϕp,‖u‖p,ϕδp}; (ii)   |Δϕδ(x)|≤1δCϕ‖∇J‖L1(Rd)(1+w1,ϕ(δ))2ϕδ(x). A.3 A ‘doubling-of-variables’ tool The following result follows along the lines of Serre (1999, Lemma 2.7.2). See also Karlsen & Storrøsten (2017, Section 6). Lemma A.9 Suppose $$u,v \in L^1_{\mathrm{loc}}({\mathbb{R}}^d)$$ and $$F$$ is Lipschitz on $${\mathbb{R}}^2$$. Fix $$\psi \in C_c({\mathbb{R}}^d)$$ and set   Tr :=∫Rd∫RdF(u(x),v(y))12dψ(x+y2)Jr(x−y2)dydx −∫RdF(u(x),v(x))ψ(x)dx, where $$J_r$$ is defined in (2.2). Then $$\mathcal{T}_r \rightarrow 0$$ as $$r \downarrow 0$$. Similarly, let $$G:[0,T] \times {\mathbb{R}} \rightarrow {\mathbb{R}}$$ be measurable in the first variable and Lipschitz continuous in the second variable. With $$w \in L^1([0,T])$$, set   Tr0(s)=∫0T|G(s,w(t))−G(s,w(s))|Jr0(t−s)dt. Then $$\mathcal{T}_{r_0}(s) \rightarrow 0$$ for a.e. $$s$$ as $$r_0 \downarrow 0$$. The above results do not rely on the symmetry of $$J$$. A.4 A version of Itô ’s formula Here we recall the particular anticipating Itô formula applied in the proof of Lemma 5.1 and Lemma A.1. The proof of this follows Nualart (2006, Theorem 3.2.2) closely. However, because of the particular assumptions, certain points simplify. See Karlsen & Storrøsten (2017, Theorem 6.7) for an outline of a proof. Theorem A.10 Let $$X$$ be a continuous process of the form   X(t)=X0+∫0tu(s)dB(s)+∫0tv(s)ds, where $$u:[0,T] \times {\it {\Omega}} \rightarrow {\mathbb{R}}$$ and $$v:[0,T] \times {\it {\Omega}} \rightarrow {\mathbb{R}}$$ are predictable processes, satisfying   E[(∫0Tu2(s,z)ds)2<∞,E[∫0Tv2(s)ds]]<∞, and $$X_0 \in L^2({\it {\Omega}},{\mathscr{F}}_0,P)$$. Let $$F:{\mathbb{R}}^2 \times [0,T] \rightarrow {\mathbb{R}}$$ be twice continuously differentiable. Suppose there exists a constant $$C > 0$$ such that for all $$(\zeta,\lambda,t) \in {\mathbb{R}}^2 \times [0,T]$$,   |F(ζ,λ,t)|,|∂3F(ζ,λ,t)|≤C(1+|ζ|+|λ|),|∂1F(ζ,λ,t)|,|∂1,22F(ζ,λ,t)|,|∂12F(ζ,λ,t)|≤C. Let $$V \in {\mathcal{S}}$$. Then $$s \mapsto \partial_1F(X(s),V,s)u(s)$$ is Skorohod integrable and   F(X(t),V,t) =F(X0,V,0) +∫0t∂3F(X(s),V,s)ds +∫0t∂1F(X(s),V,s)u(s,z)dB(s) +∫0t∂1F(X(s),V,s)v(s)ds +∫0t∂1,22F(X(s),V,s)DsVu(s)ds +12∫0t∂12F(X(s),V,s)u2(s)ds,dP-almost surely, where the stochastic integral is interpreted as a Skorohod integral. A.5 Young measures The purpose of this section is to provide a reference for some results concerning Young measures and their use in representation formulas for weak limits. For a more general introduction (see, e.g., (see, e.g., Valadier, 1994; Málek et al., 1996; Florescu & Godet-Thobie, 2012). Let $$(X,\mathscr{A},\mu)$$ be a $$\sigma$$-finite measure space and $$\mathscr{P}({\mathbb{R}})$$ the set of probability measures on $${\mathbb{R}}$$. In this article, $$X$$ is typically $${\it {\Pi}}_T \times {\it {\Omega}}$$. A Young measure from $$X$$ into $${\mathbb{R}}$$ is a function $$\nu:X \rightarrow \mathscr{P}({\mathbb{R}})$$ such that $$x \mapsto \nu_x(B)$$ is $$\mathscr{A}$$-measurable for every Borel measurable set $$B \subset {\mathbb{R}}$$. We denote by $$\mathcal{Y}\left({X,\mathscr{A},\mu;{\mathbb{R}}}\right)$$, or $$\mathcal{Y}\left({X;{\mathbb{R}}}\right)$$ if the measure space is understood, the set of all Young measures from $$X$$ into $${\mathbb{R}}$$. The following theorem is proved in Pedregal (1997, Theorem 6.2) in the case that $$X \subset {\mathbb{R}}^n$$ and $$\mu$$ is the Lebesgue measure. Theorem A.11 Fix a $$\sigma$$-finite measure space $$(X,\mathscr{A},\mu)$$. Let $$\zeta:[0,\infty) \rightarrow [0,\infty]$$ be a continuous, nondecreasing function satisfying $$\lim_{\xi \rightarrow \infty}\zeta(\xi) = \infty$$ and $$\left\{{u^n}\right\}_{n \ge 1}$$ a sequence of measurable functions such that   supn∫Xζ(|un|)dμ(x)<∞. Then there exist a subsequence $$\left\{{u^{n_j}}\right\}_{j \ge 1}$$ and $$\nu \in \mathcal{Y}\left({X,\mathscr{A},\mu;{\mathbb{R}}}\right)$$ such that for any Carathéodory function $$\psi:{\mathbb{R}} \times X \rightarrow {\mathbb{R}}$$ with $$\psi(u^{n_j}(\cdot),\cdot) \rightharpoonup \overline{\psi}$$ in $$L^1(X)$$, we have   ψ¯(x)=∫Rψ(ξ,x)dνx(ξ). The proof is based on the embedding of $$\mathcal{Y}\left({X;{\mathbb{R}}}\right)$$ into $$L^\infty_{w*}(X,{\mathbb{R}}ad{{\mathbb{R}}})$$. Here $${\mathbb{R}}ad{{\mathbb{R}}}$$ denotes the space of Radon measures on $${\mathbb{R}}$$. The crucial observation is that $$(L^1(X,C_0({\mathbb{R}})))^*$$ is isometrically isomorphic to $$L^\infty_{w*}(X,{\mathbb{R}}ad{{\mathbb{R}}})$$ also in the case that $$(X,\mathscr{A},\mu)$$ is an abstract $$\sigma$$-finite measure space. It is relatively straightforward to go through the proof and extend it to the more general case (Málek et al., 1996, Theorem 2.11). The result then follows as an application of Alaoglu’s theorem combined with the Eberlein-Šmulian theorem. Note, however, because of our use of weighted $$L^p$$-spaces, it suffices to use the version for finite measure spaces. A.6 Weak compactness in $$L^1$$. To apply Theorem A.11 it is necessary to know whether $$\left\{{\psi(\cdot,u^n(\cdot))}\right\}_{n \ge 1}$$ has a subsequence converging weakly in $$L^1(X)$$. The key result is the well-known Dunford–Pettis theorem. Definition A.12 Let $$\mathcal{K} \subset L^1(X,\mathscr{A},\mu)$$. (i) $$\mathcal{K}$$ is uniformly integrable if for any $$\varepsilon > 0$$ there exists $$c_0(\varepsilon)$$ such that   supf∈K∫|f|≥c|f|dμ≤ε whenever c≥c0(ε). (ii) $$\mathcal{K}$$ has a uniform tail if for any $$\varepsilon > 0$$ there exists $$E \in \mathscr{A}$$ with $$\mu(E) < \infty$$ such that   supf∈K∫X∖E|f|dμ≤ε. If $$\mathcal{K}$$ satisfies both (i) and (ii) it is said to be equiintegrable. Remark A.13 Note that (ii) is void when $$\mu$$ is finite. Theorem A.14 (Dunford–Pettis) Let $$(X,\mathscr{A},\mu)$$ be a $$\sigma$$-finite measure space. A subset $$\mathcal{K}$$ of $$L^1(X)$$ is relatively weakly sequentially compact if and only if it is equiintegrable. There are a couple of well-known reformulations of uniform integrability. Lemma A.15 Suppose $$\mathcal{K} \subset L^1(X)$$ is bounded. Then $$\mathcal{K}$$ is uniformly integrable if and only if (i) for any $$\varepsilon > 0$$ there exists $$\delta(\varepsilon) > 0$$ such that   supf∈K∫E|f|dμ≤ε whenever μ(E)≤δ(ε); (ii) there is an increasing function $$\Psi:[0,\infty) \rightarrow [0,\infty)$$ such that $$\Psi(\zeta)/\zeta \rightarrow \infty$$ as $$\zeta \rightarrow \infty$$ and   supf∈K∫XΨ(|f(x)|)dμ(x)<∞. Acknowledgements We are grateful to N. H. Risebro and an anonymous referee for many valuable comments. Funding Research Council of Norway through the project Stochastic Conservation Laws (250674/F20). References Bauzet C. ( 2015) Time-splitting approximation of the Cauchy problem for a stochastic conservation law. Math. Comput. Simulat. , 118, 73– 86. Google Scholar CrossRef Search ADS   Bauzet C., Charrier J. & Gallouët T. ( 2016) Convergence of flux-splitting finite volume schemes for hyperbolic scalar conservation laws with a multiplicative stochastic perturbation. Math. Comp. , 85, 2777– 2813. 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Published: Jan 1, 2018

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