The Nonlocal Inverse Problem of Donsker and Varadhan

Abstract

In this paper we prove a nonlocal version of the celebrated Inverse Problem of Donsker and Varadhan~\cite{DV} for nonlocal elliptic operators of the form $$ \cL u = L_K u + \cB_K (h,u), $$ where $L_K$ is a uniformly elliptic nonlocal operator with smooth coefficients, and, for $h:{\mathbb{R}}^N\to \mathbb{R}$ smooth and bounded, $\cB_K(h,u)$ is the associated bilinear form, which can be regarded as a nonlocal transport term.