Further study on Hunt's hypothesis (H) for Levy processes

TL;DR

This paper advances the understanding of Hunt's hypothesis (H) for Levy processes by providing new necessary and sufficient conditions, explicit examples, and decomposition results, thereby addressing a long-standing open problem in potential theory.

Contribution

It introduces two new criteria for (H), constructs Levy processes satisfying (H) in novel contexts, and shows decompositions of pure jump subordinators into parts satisfying (H).

Findings

New necessary and sufficient conditions for (H)

Explicit constructions of Levy processes satisfying (H)

Decomposition of pure jump subordinators into parts satisfying (H)

Abstract

Getoor's conjecture that essentially all Levy processes satisfy (H) is a long-standing open problem in potential theory. In the beginning of the paper, we summarize the main results obtained so far for the problem. Then, we present two new necessary and sufficient conditions for the validity of (H). Furthermore, we give applications of these new criteria. First, we give explicit constructions of Levy processes satisfying (H) in a context where previously known results could not be applied. Second, we show that a large class of pure jump subordinators can be decomposed into the summation of two independent subordinators such that both of them satisfy (H).