A large deviations principle for small perturbations of random evolution equations
The paper deals with random evolutions \(\{x^{\epsilon}(t)\), \(\epsilon >0\}\) governed by \[ dx^{\epsilon}(t)=b(x^{\epsilon}(t),y(t))dt+\sqrt{\epsilon}\sigma (x^{\epsilon}(t))dw(t). \] The random process y(t) is supposed to be independent of the d-dimensional Brownian motion w(t) and supp y\(=L^ 1([0,T],K)\) where \(K\subset {\mathbb{R}}^ m\) is compact. Then the family \(\{x^{\epsilon}(t)\), \(\epsilon >0\}\) is shown to obey a large deviations principle with the rate \[ I(g)=(1/2)\int^{T}_{0}\inf_{y\in K}| \sigma^{-1}(g(t))(\dot g(t)-b(g(t),y))|^ 2dt. \] For the case when y is a time-homogeneous n-state Markov process the author derives a Ventsel-Freidlin's type result about the asymptotical behavior as \(\epsilon\to 0\) of exit distributions of \(x^{\epsilon}(t)\) from a bounded domain.
scientific article; zbMATH DE number 4009430