New results on Hunt's hypothesis (H) for L\'{e}vy processes

TL;DR

This paper advances understanding of Hunt's hypothesis (H) for Lévy processes by establishing new conditions, demonstrating the effect of jumps, and providing examples and counterexamples of processes satisfying or not satisfying (H).

Contribution

It introduces a new necessary and sufficient condition for (H), extends the Kanda-Forst-Rao theorem, and constructs subordinators that do not meet Rao's condition.

Findings

Big jumps do not affect (H) validity.

New class of Lévy processes satisfying (H).

Subordinators not satisfying Rao's condition.

Abstract

In this paper, we present new results on Hunt's hypothesis (H) for L\'{e}vy processes. We start with a comparison result on L\'{e}vy processes which implies that big jumps have no effect on the validity of (H). Based on this result and the Kanda-Forst-Rao theorem, we give examples of subordinators satisfying (H). Afterwards we give a new necessary and sufficient condition for (H) and obtain an extended Kanda-Forst-Rao theorem. By virtue of this theorem, we give a new class of L\'{e}vy processes satisfying (H). Finally, we construct a type of subordinators that does not satisfy Rao's condition.