Hunt's hypothesis (H) and Getoor's conjecture for L\'{e}vy Processes

TL;DR

This paper investigates Hunt's hypothesis (H) and Getoor's conjecture for Lévy processes, establishing conditions under which (H) holds, including non-degeneracy of the Lévy-Khintchine exponent and specific integral equations, with implications for subordinators.

Contribution

The paper provides new criteria linking the Lévy-Khintchine parameters to Hunt's hypothesis (H), including explicit conditions involving the Lévy measure and drift, and characterizes subordinators satisfying (H).

Findings

If A is non-degenerate, then X satisfies (H).

X satisfies (H) iff a certain integral equation has a solution under specified conditions.

Subordinators satisfying (H) must have zero drift coefficient.

Abstract

In this paper, Hunt's hypothesis (H) and Getoor's conjecture for L\'{e}vy processes are revisited. Let $X$ be a L\'{e}vy process on $\mathbf{R}^n$ with L\'{e}vy-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.}