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

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.