Resumen de Averaging principle of SDE with small diffusion: Moderate deviations

A. Guillin

• Consider the following stochastic differential equation in Rd:

  dXεtXε0==b(Xεt,ξt/ε)dt+ε√a(Xεt,ξt/ε)dWt,x0, where the random environment (ξt) is an exponentially ergodic Markov process, independent of the Wiener process (Wt), with invariant probability measure π, and ε is some small parameter. In this paper we prove the moderate deviations for the averaging principle of Xε, that is, deviations of (Xεt) around its limit averaging system (x¯t) given by %d¯. % dx¯t=b¯(x¯t)dt and x¯0=x0 where b¯(x)=Eπ(b(x,⋅)). More precisely we obtain the large deviation estimation about (ηεt=Xεt−x¯tε√h(ε))t∈[0,1] in the space of continuous trajectories, as ε decreases to 0, where h(ε) is some deviation scale satisfying 1≪h(ε)≪ε−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.