Stochastic integral representations for the Ray-Knight theorem of the Levy forest

TL;DR

This paper introduces a stochastic integral representation for local times of height processes in spectrally positive Levy processes, deriving a strong stochastic equation and extending Ray-Knight theorems to genealogical forests of continuous-state branching processes.

Contribution

It provides a novel stochastic integral representation for local times, leading to new formulations of Ray-Knight theorems for Levy processes and their genealogical structures.

Findings

Derived a stochastic integral representation for local times.

Established a strong stochastic equation for local time processes.

Extended Ray-Knight theorems to Levy process genealogies.

Abstract

We present a simple stochastic integral representation for the local times of the height process of a spectrally positive Levy process stopped at a hitting time. From the representation we derive a strong stochastic equation for the local time process of the type of Bertoin and Le Gall (Illinois J. Math., 2006) and Dawson and Li (Ann. Probab., 2012). This leads to a representation of the Ray-Knight theorem of Le Gall and Le Jan (Ann. Probab., 1998) and Duquesne and Le Gall (Asterisque, 2002), which codes the genealogical forest of a continuous-state branching process. The results extend those in the recent work of Aidekon et al. (Sci. China Math., 2024) for a Brownian motion with a local time drift.