Averaging principle of SDE with small diffusion: Moderate deviations

• 成果类型: Article
• 署名作者: Guillin, A
• 署名单位: Universite PSL; Universite Paris-Dauphine
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2003
• 页码: 413-443
• 关键词: STOCHASTIC DIFFERENTIAL-EQUATIONS; MARKOV-PROCESSES; dependence; systems

摘要

Consider the following stochastic differential equation in R-d: dX(t)(epsilon) = b(X-t(epsilon), xi(t/epsilon)) dt + rootepsilona(X-t(epsilon), xi(t/epsilon)) dW(t), X-0(epsilon) = X-0, where the random environment (xi(t)) is an exponentially ergodic Markov process, independent of the Wiener process (W-t), with invariant probability measure pi, and epsilon is some small parameter. In this paper we prove the moderate deviations for the averaging principle of X-epsilon, that is, deviations of (X-t(epsilon)) around its limit averaging system ((x) over bar (t)) given by d (x) over bar (t) = (b) over bar((x) over bar (t))dt and (x) over bar (0) = x(0) where (b) over bar (x) = E-pi(b(x, .)). More precisely we obtain the large deviation estimation about (eta(t)(epsilon) = X-t(epsilon) - rootepsilonh(epsilon)/(x) over bar (t))(tepsilon[0, 1]) in the space of continuous trajectories, as epsilon decreases to 0, where h(epsilon) is some deviation scale satisfying 1 much less than h(epsilon) much less than epsilon(-1/2). Our strategy will be first to show the exponential tightness and then the local moderate deviation principle, which relies on some new method involving a conditional Schilder's theorem and a moderate deviation principle for inhomogeneous integral functionals of Markov processes, previously established by the author in Guillin (2001).