Uniform large deviations for parabolic SPDEs and applications

Let C-z([0,T] X [0,1]) denote the set of functions f(t,x) which are alpha-Holder continuous in t and 2 alpha-Holder continuous in x. For 0<alpha<1/4 we prove a large deviation principle in a separable subset of C-alpha([0,T] x [0,1]) for the solution X-phi(epsilon)(t,x) to a parabolic stochastic partial differential equation perturbed by a small non-linear white noise, uniformly when the initial condition phi belongs to a compact subset of C-2 alpha,C-0([0,1]). This does not require any boundedness or nondegeneracy on the coefficients, and is applied to deduce asymptotics for the exit time of X-phi(epsilon)(t,.) from a bounded domain of C C-2 alpha,C-0([0,1]). (C) 1997 Elsevier Science B.V.