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 $(a,A,\mu )$. {First, we show that if $A$ is non-degenerate then $X$ satisfies (H). Second, under the assumption that $\mu ({\mathbf{R}}^n\backslash \sqrt{A}{\mathbf{R}}^n)<\infty$, we show that $X$ satisfies (H) if and only if the equation $$\sqrt{A}y=-a-{\int }_{\{x\in {\mathbf{R}}^n\backslash \sqrt{A}{\mathbf{R}}^n:|x|<1\}}x\mu (dx),y\in {\mathbf{R}}^n,$$ has at least one solution. Finally, we show that if $X$ is a subordinator and satisfies (H) then its drift coefficient must be 0.}

11 pages. arXiv admin note: text overlap with arXiv:1210.2016

References in corpus (1)

• A note on the existence of transition probability densities for Lévy processes

Cited by in corpus (6)

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• Levy-Khintchine type representation of Dirichlet generators and Semi-Dirichlet forms
• Hunt's Hypothesis (H) for Markov Processes: Survey and Beyond