Uniform large deviations for parabolic SPDEs and applications

Let denote the set of functions f(t,x) which are [alpha]-Hölder continuous in t and 2[alpha]-Hölder continuous in x. For 0 < [alpha] < 1/4 we prove a large deviation principle in a separable subset of for the solution X[var epsilon][phi](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 . This does not require any boundedness or non-degeneracy on the coefficients, and is applied to deduce asymptotics for the exit time of X[var epsilon][phi](t,1) from a bounded domain of .