On the Cauchy problem for integro-differential operators in Sobolev classes and the martingale problem

TL;DR

This paper establishes existence and uniqueness of solutions in Sobolev spaces for a class of parabolic integro-differential equations of order alpha in (0,2), and applies these results to the martingale problem.

Contribution

It provides new results on well-posedness of integro-differential equations with nonlocal operators in Sobolev spaces, including applications to stochastic processes.

Findings

Existence and uniqueness of solutions in Sobolev spaces for the Cauchy problem.

Application of results to prove well-posedness of the martingale problem.

Analysis of operators with kernels having bounded, nondegenerate, and Hölder continuous properties.

Abstract

The existence and uniqueness in Sobolev spaces of solutions of the Cauchy problem to parabolic integro-differential equation of the order {\alpha}\in(0,2) is investigated. The principal part of the operator has kernel m(t,x,y)/|y|^{d+{\alpha}} with a bounded nondegenerate m, H\"older in x and measurable in y. The lower order part has bounded and measurable coefficients. The result is applied to prove the existence and uniqueness of the corresponding martingale problem.