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

TL;DR

This paper derives large deviation estimates for solutions of certain non-local linear parabolic equations involving Lévy measures, providing bounds on convergence rates based on the decay properties of the measure.

Contribution

It establishes exponential convergence rates for solutions in bounded domains to the whole space solution under minimal assumptions on the Lévy measure.

Findings

Solutions converge exponentially fast to the whole space solution.

The convergence rate depends on the decay of the Lévy measure at infinity.

Provided bounds are applicable even when the Lévy measure is singular at the origin.

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 L\'evy 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.