An infinite-dimensional representation of the Ray-Knight theorems

TL;DR

This paper extends the classical Ray-Knight theorems by describing the joint local time process for all levels using stochastic integrals with white noise, linking height processes of CSBPs with stochastic differential equations.

Contribution

It introduces an infinite-dimensional representation of the Ray-Knight theorems applicable to $mbda$-processes, connecting local times with white noise-driven SDEs for CSBPs with immigration.

Findings

Provides an explicit relation between local times and white noise integrals.

Shows the stochastic differential equation is a reformulation of Tanaka's formula.

Extends Ray-Knight theorems to a joint, infinite-dimensional setting.

Abstract

The classical Ray-Knight theorems for Brownian motion determine the law of its local time process either at the first hitting time of a given value a by the local time at the origin, or at the first hitting time of a given position b by Brownian motion. We extend these results by describing the local time process jointly for all a and all b, by means of stochastic integral with respect to an appropriate white noise. Our result applies to $\mu$-processes, and has an immediate application: a $\mu$-process is the height process of a Feller continuous-state branching process (CSBP) with immigration (Lambert [10]), whereas a Feller CSBP with immigration satisfies a stochastic differential equation driven by a white noise (Dawson and Li [7]); our result gives an explicit relation between these two descriptions and shows that the stochastic differential equation in question is a reformulation of Tanaka's formula.