A large-deviation principle for random evolution equations

We consider the family of stochastic processes X-epsilon = {X-epsilon(t), 0 less than or equal to t less than or equal to 1}, epsilon > 0, where X-eps ilon is the solution of the It (o) over cap stochastic differential equatio n dX(epsilon)(t)= root epsilon sigma (X-epsilon(t), Z(t)) dW(t) + b(X-epsilon (t), Y(t)) dt, whose coefficients depend on processes Z(t) = {Z(t), t is an element of [0, 1]} and Y(t) = {Y(t), i is an element of [0, 1]}. Using an extended 'contr action principle', we give the large-deviation principle (LDP) of X-epsilon as epsilon --> 0. This extends the LDP for a random evolution equation, pr oved by Yi-Jun Hu, to the case of random diffusion coefficients.