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 d, 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.

Approximation of quasi-potentials and exit problems for multidimensional RDE’s with noise / Sandra Cerrai; Mark Freidlin. - In: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9947. - STAMPA. - 363:(2011), pp. 3853-3892. [10.1090/S0002-9947-2011-05352-3]

Approximation of quasi-potentials and exit problems for multidimensional RDE’s with noise

CERRAI, SANDRA;
2011

Abstract

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 d, 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.
2011
363
3853
3892
Sandra Cerrai; Mark Freidlin
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/947194
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