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

 

Large deviation estimates for some nonlocal equations. General bounds and applications


Authors: Cristina Brändle and Emmanuel Chasseigne
Journal: Trans. Amer. Math. Soc. 365 (2013), 3437-3476
MSC (2010): Primary 47G20, 60F10; Secondary 35A35, 49L25
Published electronically: February 7, 2013
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Abstract: Large deviation estimates for the following linear parabolic equation are studied:

$\displaystyle \frac {\partial u}{\partial t}=\mathop {\rm Tr}\Big ( a(x)D^2u\Big ) + b(x)\cdot D u+ \mathcal {L}[u](x), $

where $ \mathcal {L}[u]$ is a nonlocal Lévy-type term associated to a Lévy measure $ \mu $ (which may be singular at the origin):

$\displaystyle \mathcal {L}[u](x)=\int _{\mathbb{R}^N} \Big \{(u(x+y)-u(x)-(D u(x)\cdot y) 1\!\!{\rm I}_{\{\vert y\vert<1\}} (y)\Big \}\mathrm {d}\mu (y)\,. $

Assuming only that some negative exponential integrates with respect to the tail of $ \mu $, it is shown that given an initial data, solutions defined in a bounded domain converge exponentially fast to the solution of the problem defined in the whole space. The exact rate, which depends strongly on the decay of $ \mu $ at infinity, is also estimated.

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Additional Information

Cristina Brändle
Affiliation: Departamento de Matemáticas, Universidad Carlos III Madrid, Avda. de la Universidad, 30, 28911 Leganés, Spain
Email: cbrandle@math.uc3m.es

Emmanuel Chasseigne
Affiliation: Laboratoire de Mathématiques et Physique Théorique, UMR7350, Université F. Rabelais - Tours, Parc de Grandmont, 37200 Tours, France – and – Fédération de Recherche Denis Poisson - FR2964 - Université d’Orléans & Université F. Rabelais - Tours
Email: echasseigne@univ-tours.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05629-2
PII: S 0002-9947(2013)05629-2
Keywords: Nonlocal diffusion, large deviation, Hamilton-Jacobi equation, Lévy operators.
Received by editor(s): April 14, 2010
Received by editor(s) in revised form: May 12, 2011
Published electronically: February 7, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.