Given a separable and real Hilbert space $${\mathbb {H}}$$ H and a trace-class, symmetric and non-negative operator $$\mathscr {G}\,{:}\,{\mathbb {H}}\rightarrow {\mathbb {H}}$$ G : H → H , we examine the equation $$\begin{aligned} dX_t = -X_t dt + b(X_t) dt + \sqrt{2} dW_t, \quad X_0=x\in {\mathbb {H}}, \end{aligned}$$ d X t = - X t d t + b ( X t ) d t + 2 d W t , X 0 = x ∈ H , where $$(W_t)$$ ( W t ) is a $$\mathscr {G}$$ G -Wiener process on $${\mathbb {H}}$$ H and $$b\,{:}\,{\mathbb {H}}\rightarrow {\mathbb {H}}$$ b : H → H is Lipschitz. We assume there is a splitting of $${\mathbb {H}}$$ H into a finite-dimensional space $${\mathbb {H}}^l$$ H l and its orthogonal complement $${\mathbb {H}}^h$$ H h such that $$\mathscr {G}$$ G is strictly positive definite on $${\mathbb {H}}^l$$ H l and the non-linearity b admits a contraction property on $${\mathbb {H}}^h$$ H h . Assuming a geometric drift condition, we derive a Kantorovich ( $$L^1$$ L 1 Wasserstein) contraction with an explicit contraction rate for the corresponding Markov kernels. Our bounds on the rate are based on the eigenvalues of $$\mathscr {G}$$ G on the space $${\mathbb {H}}^l$$ H l , a Lipschitz bound on b and a geometric drift condition. The results are derived using coupling methods.
Stochastical Partial Differential Equations – Springer Journals
Published: Jan 30, 2017
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