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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Published electronically: February 10, 2011
MathSciNet review: 2775830
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Abstract: We deal with a class of reaction-diffusion equations, in space dimension $ d>1$, perturbed by a Gaussian noise $ \partial w^\delta/\partial t$ which is white in time and colored in space. We assume that the noise has a small correlation radius $ \delta$, so that it converges to the white noise $ \partial w/\partial t$, as $ \delta\downarrow 0$. By using arguments of $ \Gamma$-convergence, we prove that, under suitable assumptions, the quasi-potential $ V_\delta$ converges to the quasi-potential $ V$, 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: http://dx.doi.org/10.1090/S0002-9947-2011-05352-3
PII: S 0002-9947(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