A large-deviation principle for random evolution equations

Abstract

We consider the family of stochastic processes X^ε= {X^ε(t), 0≤t≤1}, ε>0, where X^ε is the solution of the Itô stochastic differential equation </p><p> //\rm d}X^ε(t)=\sqrt{ε}σ(X^ε(t),Z(t))\rm d}W_t+b(X^ε(t),Y(t))\rm d}t,// </p><p> whose coefficients depend on processes Z(t)= {Z(t),t∈[0,1]} and Y(t)={Y(t),t∈[0,1]}. Using an extended `contraction principle', we give the large-deviation principle (LDP) of X^ε as ε→0. This extends the LDP for a random evolution equation, proved by Yi-Jun Hu, to the case of random diffusion coefficients.