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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Approximation of quasi-potentials and exit problems for multidimensional RDE’s with noise
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by Sandra Cerrai and Mark Freidlin PDF
Trans. Amer. Math. Soc. 363 (2011), 3853-3892 Request permission

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
  • 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
  • © Copyright 2011 American Mathematical Society
  • 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
  • MathSciNet review: 2775830