An inequality for a class of Markov 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 $n(x,h)$.