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

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.