A unified approach to the large deviations for small perturbations of random evolution equations

LetXɛ = {Xɛ (t ; 0 ⩽t ⩽ 1 } (ɛ > 0) be the processes governed by the following stochastic differential equations:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$dX^\varepsilon (t) = \sqrt \varepsilon \sigma (X^\varepsilon (t))dB(t) + b(X^\varepsilon (t),\nu (t))dt,$$ \end{document} wherev(t) is a random process independent of the Brownian motionB(·). Some large deviation (LD) properties of { (Xɛ, ν(.)); ɛ > 0} are proved. For a particular case, an explicit representation of the rate function is also given, which solves a problem posed by Eizenberg and Freidlin. In the meantime, an abstract LD theorem is obtained.