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

publication date

  • July 2013

start page

  • 3437

end page

  • 3476

issue

  • PII S0002-9947(2013)05629

volume

  • 365

International Standard Serial Number (ISSN)

  • 0002-9947

abstract

  • Large deviation estimates for the following linear parabolic equation are studied: partial derivative u/partial derivative t = Tr(a(x)D(2)u) + b(x) . Du + L[u](x), where L[u] is a nonlocal Levy-type term associated to a Levy measure mu (which may be singular at the origin): L[u](x) = integral(RN) {(u(x + y) - u(x) - (Du(x) . y)I-{vertical bar y vertical bar<1}(y)}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.

keywords

  • nonlocal diffusion; large deviation; hamilton-jacobi equation; levy operators