Uniform large deviations for parabolic SPDEs and applications

Let Cα([0, T] × [0, 1]) denote the set of functions f(t,x) which are α-Hölder continuous in t and 2α-Hölder continuous in x. For 0 < α < 14 we prove a large deviation principle in a separable subset of Cα([0, T] × [0, 1]) for the solution Xεϕ(t,x) to a parabolic stochastic partial differential equation perturbed by a small non-linear white noise, uniformly when the initial condition ϕ belongs to a compact subset of C2α,0([0, 1]). This does not require any boundedness or non-degeneracy on the coefficients, and is applied to deduce asymptotics for the exit time of Xεϕ(t,1) from a bounded domain of C2α,0([0, 1]).