Hunt's hypothesis (H) and Getoor's conjecture for Lévy Processes

Abstract

In this paper, Hunt's hypothesis (H) and Getoor's conjecture for Lévy processes are revisited. Let $X$ be a Lévy process on $\mathbf{R}^n$ with Lévy-Khintchine exponent $(a,A,μ)$. {First, we show that if $A$ is non-degenerate then $X$ satisfies (H). Second, under the assumption that $μ({\mathbf{R}^n\backslash \sqrt{A}\mathbf{R}^n})<\infty$, we show that $X$ satisfies (H) if and only if the equation $$ \sqrt{A}y=-a-\int_{\{x\in {\mathbf{R}^n\backslash \sqrt{A}\mathbf{R}^n}:\,|x|<1\}}xμ(dx),\ y\in \mathbf{R}^n, $$ has at least one solution. Finally, we show that if $X$ is a subordinator and satisfies (H) then its drift coefficient must be 0.}