An inequality for a class of Markov processes

TL;DR

This paper establishes a Krylov-type inequality for a class of Markov processes generated by a non-local operator with variable kernels, enabling existence results for associated martingale problems without kernel continuity assumptions.

Contribution

It introduces a Krylov-type inequality for Markov processes with variable kernels and proves existence of solutions to the martingale problem without requiring kernel continuity.

Findings

Established a Krylov-type inequality for the operator  involving variable kernels.

Proved existence of solutions to the martingale problem without kernel continuity.

Extended the theory of non-local operators and associated stochastic processes.

Abstract

Let $\alpha \in (0,2)$ and consider the operator $\sL$ given by \[ \sL f(x)=\int[ f(x+h)-f(x)-1_{(|h|\leq 1)}h\cdot \grad f(x)]\frac{n(x,h)}{|h|^{d+\alpha}} \d h, \] where the term $1_{(|h|\leq 1)}h\cdot \grad f(x)$ is not present when $\alpha \in (0,1)$. Under some suitable assumptions on the kernel $n(x,h)$, we prove a Krylov-type inequality for processes associated with $\sL$. As an application of the inequality, we prove the existence of a solution to the martingale problem for $\sL$ without assuming any continuity of $n(x,h)$.