Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems.
type: Item
instance of: scholarly article
title: Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems.
zbMATH Open document ID: 1047.60059
DOI: 10.1016/S0304-4149(00)00061-2
published in: Stochastic Processes and their Applications
publication date: 2004-09-22T00:00:00Z
review text: The paper is devoted to an in-depth study of long-time behaviour of solutions to a stochastic Hamiltonian system with damping \[ dx_ {t} = y_ {t}\, dt, \quad dy_ {t} = -c(x_ {t},y_ {t})y_ {t} \,dt -\nabla V(x_ {t})\, dt + \Sigma (x_ {t},y_ {t})\,dW_ {t} \tag{1} \] in \(\mathbb R^ {2d}\), where \(W\) is a standard \(d\)-dimensional Wiener process. It is supposed that the potential \(V\in C^ 1(\mathbb R^ {d})\) is lower bounded, the damping coefficient \(c\) is continuous and bounded on every set \(\{\| x\| \leq N\}\times \mathbb R^ {d}\), \(N>0\), and the matrix-valued function \(\Sigma \) is \(C^ \infty \)-smooth and \(0<\Sigma \leq \sigma I\) on \(\mathbb R^ {2d}\) for some \(\sigma >0\). Moreover, it it assumed that there exist \(c,L>0\) such that \(c^ {s}(x,y)\geq cI\) for all \(\| x\| \geq L\), \(y\in \mathbb R^ {d}\), \(c^ {s} = \frac 12(c+c^ {T})\) denoting the symmetrization of the matrix \(c\). First, it is shown that there exists a unique weak solution to (1) for every initial datum \((x_ 0,y_ 0)\in \mathbb R^ {2d}\). A Girsanov formula is established and used to prove that the transition semigroup \((P_ {t})\) defined by (1) is strong Feller in the strict sense. Further, several general theorems on sufficient conditions for (empirical measures of) strong Feller topologically transitive Markov processes on Polish spaces to obey large or moderate deviation principles are proven. Finally, these results are applied to solutions of the equation (1). In particular, assume that there exist a function \(G\in C^ 1 _ {b}(\mathbb R^ {d};\mathbb R^ {d})\), \(\| G\| \leq 1\) on \(\mathbb R^ {d}\), and a lower bounded function \(U\in C^ 1(\mathbb R^ {d})\) satisfying \[ \begin{aligned} & \liminf _ {\| x\| \to \infty } \nabla V(x)\cdot G(x)>0,\;\lim _ {\| x\| \to \infty } \| (\partial _ {x_ {i}} G_ {j}(x))\| = 0,\\ & \lim _ {\| x\| \to \infty } \sup _ {y\in \mathbb R^ {d}} \| c(x,y)^ {T}G(x) - \nabla U(x) \| = 0.\end{aligned} \] Then the transition semigroup \((P_t)\) is…
reviewed by: Jan Seidler
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60J35
zbMATH DE Number: 2102367
zbMATH Keywords: stochastic Hamiltonian systems
zbMATH Keywords: large deviations
zbMATH Keywords: moderate deviations
zbMATH Keywords: exponential convergence
zbMATH Keywords: strong Feller property
author: Liming Wu
MaRDI profile type: Publication
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full work available at URL: s0304-4149(00)00061-2
OpenAlex ID: W2046017653
description: scientific article; zbMATH DE number 2102367