# Large Deviations estimates for some non-local equations. General bounds and applications

Abstract : Large deviation estimates for the following linear parabolic equation are studied: $\frac{\partial u}{\partial t}=\tr\Big( a(x)D^2u\Big) + b(x)\cdot D u + \int_{\R^N} \Big\{(u(x+y)-u(x)-(D u(x)\cdot y)\ind{|y|<1}(y)\Big\}\d\mu(y)\,,$ where $\mu$ is a Lévy measure (which may be singular at the origin). 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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https://hal.archives-ouvertes.fr/hal-00414225
Contributor : Emmanuel Chasseigne <>
Submitted on : Tuesday, September 8, 2009 - 1:59:30 PM
Last modification on : Friday, February 19, 2021 - 4:10:02 PM
Long-term archiving on: : Wednesday, September 22, 2010 - 1:14:04 PM

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### Identifiers

• HAL Id : hal-00414225, version 2
• ARXIV : 0909.1467

### Citation

Cristina Brändle, Emmanuel Chasseigne. Large Deviations estimates for some non-local equations. General bounds and applications. Transactions of the American Mathematical Society, American Mathematical Society, 2013, 365, pp.3437-3476. ⟨hal-00414225v2⟩

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