Acta Applicandae Mathematicae

doi: 10.1023/A:1017331722403pmid: N/A

Accardi, Luigi; Lu, Yun; Volovich, Igor

doi: 10.1023/A:1010719902298pmid: N/A

During the past 15 years a new technique, called the stochastic limit of quantum theory, has been applied to deduce new, unexpected results in a variety of traditional problems of quantum physics, such as quantum electrodynamics, bosonization in higher dimensions, the emergence of the noncrossing diagrams in the Anderson model, and in the large-N-limit in QCD, interacting commutation relations, new photon statistics in strong magnetic fields, etc. These achievements required the development of a new approach to classical and quantum stochastic calculus based on white noise which has suggested a natural nonlinear extension of this calculus. The natural theoretical framework of this new approach is the white-noise calculus initiated by T. Hida as a theory of infinite-dimensional generalized functions. In this paper, we describe the main ideas of the white-noise approach to stochastic calculus and we show that, even if we limit ourselves to the first-order case (i.e. neglecting the recent developments concerning higher powers of white noise and renormalization), some nontrivial extensions of known results in classical and quantum stochastic calculus can be obtained.

Albeverio, Sergio; Daletskii, Alexei; Kondratiev, Yuri; Röckner, Michael

doi: 10.1023/A:1010736106899pmid: N/A

Stochastic dynamics associated with Gibbs measures on M Z d , where M is a compact Riemannian manifold and Z d is an integer lattice, is considered. Equivalence of its L 2-ergodicity and the extremality of the corresponding Gibbs measure is proved.

Arai, Asao

doi: 10.1023/A:1010744324646pmid: N/A

We consider arithmetical aspects of analysis on Fock spaces (Boson Fock space, Fermion Fock space, and Boson–Fermion Fock space) with applications to analytic number theory.

Asai, Nobuhiro; Kubo, Izumi; Kuo, Hui-Hsiung

doi: 10.1023/A:1010738827855pmid: N/A

Let {b k (n)} n=0 ∞ be the Bell numbers of order k. It is proved that the sequence {b k (n)/n!} n=0 ∞ is log-concave and the sequence {b k (n)} n=0 ∞ is log-convex, or equivalently, the following inequalities hold for all n⩾0, $$1 \leqslant \frac{{b_k (n + 2)b_k (n)}}{{b_k (n + 1)^2 }} \leqslant \frac{{n + 2}}{{n + 1}}$$ . Let {α(n)} n=0 ∞ be a sequence of positive numbers with α(0)=1. We show that if {α(n)} n=0 ∞ is log-convex, then α(n)α(m)⩽α(n+m), ∀n,m⩾0. On the other hand, if {α(n)/n!} n=0 ∞ is log-concave, then $$\alpha (n + m) \leqslant \left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right)\alpha (n)\alpha (m),{\text{ }}\forall n,m \geqslant 0$$ . In particular, we have the following inequalities for the Bell numbers $$b_k (n)b_k (m) \leqslant b_k (n + m) \leqslant \left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right)b_k (n)b_k (m),{\text{ }}\forall n,m \geqslant 0$$ . Then we apply these results to characterization theorems for CKS-space in white noise distribution theory.

Chung, Dong; Ji, Un

doi: 10.1023/A:1010724103207pmid: N/A

In this paper we shall show the heredity of a differentiable one-parameter semigroup under the second quantization and then discuss the resolvent of the differential second quantization operator and the potentials of test white noise functionals. As an application, we shall investigate the existence of solutions of the Poisson-type equations associated with differential second quantization operators as well as operators similar to differential second quantization operators.

Dôku, Isamu

doi: 10.1023/A:1010747031490pmid: N/A

Nonlinear equation with catalytic noise is considered. We discuss the existence of catalytic superprocess associated with the equation and derive the exponential moment formula. Moreover, we prove the large deviation principle for catalytic superprocesses.

Ezawa, Hiroshi

doi: 10.1023/A:1010780220954pmid: N/A

Introducing a path integral for the Ornstein–Uhlenbeck process distorted by a potential V(x), we find out the T→∞ limit of the probability distributions of X[ω]:=1/T ν∫0 T V(ω(t)) dt for Ornstein–Uhlenbeck process ω(t), with appropriate values of the exponent ν that depend on V. The results are compared with those for the Wiener process.

Hibino, Yuji; Hitsuda, Masuyuki; Muraoka, Hiroshi

doi: 10.1023/A:1010782826889pmid: N/A

For a stationary centered Gaussian process, we construct a noncanonical representation which has an infinite-dimensional orthogonal complement that is nontrivial. The authors have already proposed a systematic method for the construction of noncanonical representation having a finite-dimensional orthogonal complement.

Holden, Helge; Øksendal, Bernt

doi: 10.1023/A:1010730510108pmid: N/A

We illustrate the use of white noise analysis in the solution of stochastic partial differential equations by explicitly solving the stochastic Neumann boundary-value problem LU(x)−c(x)U(x)=0, x∈D⊂R d ,γ(x)⋅∇U(x)=−W(x), x∈∂D, where L is a uniformly elliptic linear partial differential operator and W(x), x∈R d , is d-parameter white noise.