## 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