Approximation of Quasi-potentials and Exit Problems for Multidimensional Rde’s with Noise

We deal with a class of reaction-diffusion equations, in space dimension d > 1, perturbed by a Gaussian noise ∂wδ/∂t which is white in time and colored in space. We assume that the noise has a small correlation radius δ, so that it converges to the white noise ∂w/∂t, as δ ↓ 0. By using arguments of Γ-convergence, we prove that, under suitable assumptions, the quasi-potential Vδ converges to the quasi-potential V , corresponding to spacetime white noise, in spite of the fact that the equation perturbed by space-time white noise has no solution. We apply these results to the asymptotic estimate of the mean of the exit time of the solution of the stochastic problem from a basin of attraction of an asymptotically stable point for the unperturbed problem.