Large deviation estimates for some nonlocal equations. General bounds and applications
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- by Cristina Brändle and Emmanuel Chasseigne PDF
- Trans. Amer. Math. Soc. 365 (2013), 3437-3476 Request permission
Abstract:
Large deviation estimates for the following linear parabolic equation are studied: \[ \frac {\partial u}{\partial t}=\textrm {Tr}\Big ( a(x)D^2u\Big ) + b(x)\cdot D u+ \mathcal {L}[u](x), \] where $\mathcal {L}[u]$ is a nonlocal Lévy-type term associated to a Lévy measure $\mu$ (which may be singular at the origin): \[ \mathcal {L}[u](x)=\int _{\mathbb {R}^N} \Big \{(u(x+y)-u(x)-(D u(x)\cdot y) 1\!\!\textrm {I}_{\{|y|<1\}} (y)\Big \}\mathrm {d}\mu (y) . \] 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.References
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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
- Received by editor(s): April 14, 2010
- Received by editor(s) in revised form: May 12, 2011
- Published electronically: February 7, 2013
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 3437-3476
- MSC (2010): Primary 47G20, 60F10; Secondary 35A35, 49L25
- DOI: https://doi.org/10.1090/S0002-9947-2013-05629-2
- MathSciNet review: 3042591