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

Authors:
Sandra Cerrai and Mark Freidlin

Journal:
Trans. Amer. Math. Soc. **363** (2011), 3853-3892

MSC (2010):
Primary 60H15, 60F10, 35K57, 49J45

DOI:
https://doi.org/10.1090/S0002-9947-2011-05352-3

Published electronically:
February 10, 2011

MathSciNet review:
2775830

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Abstract | References | Similar Articles | Additional Information

Abstract: We deal with a class of reaction-diffusion equations, in space dimension , perturbed by a Gaussian noise which is white in time and colored in space. We assume that the noise has a small correlation radius , so that it converges to the white noise , as . By using arguments of -convergence, we prove that, under suitable assumptions, the quasi-potential converges to the quasi-potential , corresponding to space-time 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.

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Additional Information

**Sandra Cerrai**

Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742

**Mark Freidlin**

Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742

DOI:
https://doi.org/10.1090/S0002-9947-2011-05352-3

Received by editor(s):
June 5, 2009

Received by editor(s) in revised form:
March 19, 2010

Published electronically:
February 10, 2011

Additional Notes:
The second author was partially supported by an NSF grant

Article copyright:
© Copyright 2011
American Mathematical Society