Existence of a Strong Solution for an Integro-Differential Equation and Superposition of Diffusion Processes

Abstract

The purpose of this article is to show the existence and the uniqueness of a strong solution $u \in {\widehat C^{2,\alpha }}$ to an integro-differential equation $\left\{ {\mu - \left( {A + B} \right)} \right\}u = f$ for each μ > 0 and $f \in {\widehat C^\alpha }$, where A is a second order elliptic differential operator, B a Lévy type integral operator, ${\widehat C^{k,\alpha }}$ the space of all k-times continuously differentiable functions with α-Hölder continuous k-th derivatives and with all j-th derivatives for j ≤ k vanishing at infinity, and ${\widehat C^\alpha } = {\widehat C^{0,\alpha }}$. This ensures the existence of a Feller process associated with the generator A + B in the usual way.