Ray-Knight Theorems for Spectrally Negative L\'evy Processes

TL;DR

This paper extends Ray-Knight theorems to spectrally negative Lévy processes, describing local time processes at specific stopping times, their branching structures, and their non-Markovian dependencies due to jumps.

Contribution

It provides a novel generalization of Ray-Knight theorems for spectrally negative Lévy processes, including descriptions of local times, their Lévy measures, and excursion-based constructions.

Findings

Local times are infinitely divisible with explicit Lévy measures.

Branching structures of local times are characterized using excursion theory.

Local times exhibit non-Markovian dependencies due to jumps.

Abstract

In this paper, we study the law of the local time processes $(L_T^x(X),x\in \mathbb{R})$ associated to a spectrally negative L\'evy process $X$, in the cases $T=\tau_a^+$, the first passage time of $X$ above $a>0$ and $T=\tau(c)$, the first time it accumulates $c$ units of local time at zero. We describe the branching structure of local times and Poissonian constructions of them using excursion theory. The presence of jumps for $X$ creates a type of excursions which can contribute simultaneously to local times of levels above and below a given reference point. This fact introduces dependency on local times, causing them to be non-Markovian. Nonetheless, the overshoots and undershoots of excursions will be useful to analyze this dependency. In both cases, local times are infinitely divisible and we give a description of the corresponding L\'evy measures in terms of excursion measures. These are hence analogues in the spectrally negative L\'evy case of the first and second Ray-Knight theorems, originally stated for the Brownian motion.