A Large Deviations Principle for Small Perturbations of Random Evolution Equations

Abstract

We prove a result for small random perturbations of random evolution equations analogous to the Ventsel-Freidlin result on small perturbations of dynamical systems. In particular, we derive large deviations estimates and indicate how they can be used to prove an exit result. The processes we study are governed by equations of the form $dx^\varepsilon(t) = b(x^\varepsilon(t), y(t)) dt + \sqrt \varepsilon \sigma(x^\varepsilon(t)) dw(t)$, where $x^0$ is already a random process. The results include the case where $y$ is an $n$-state Markov process. In the special case $\sigma \equiv Id$, the proof of the estimates is a consequence of a generalization of the "contraction principle" for large deviations: We give sufficient conditions on a continuous function $F$, which ensure that if $\{X_\varepsilon: \varepsilon > 0\}$ satisfies a large deviations principle, then so does $\{F(X_\varepsilon, Y): \varepsilon > 0\}$, where $Y$ is independent of $\{X_\varepsilon: \varepsilon > 0\}$.