Integro-differential equations with L{\'e}vy operators for degenerate jumps depending on spaces and gradients

TL;DR

This paper proves the comparison principle and existence of solutions for a class of integro-differential equations involving Lévy operators, which model jump processes that can be degenerate and depend on space and gradient.

Contribution

It introduces new results on the well-posedness of degenerate Lévy-driven integro-differential equations with gradient dependence.

Findings

Established comparison principle for the equations

Proved existence of solutions under degeneracy and gradient dependence

Extended theory to include space- and gradient-dependent Lévy operators

Abstract

We establish the comparison principle and the existence of solutions of the integro-differential equations with L{\'e}vy operators. The L{\'e}vy operators of our interest are infinitesmal generator of the jump processes which could be degenerate and may depend on the gradient of the solutions.