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Journal Articles Science China Mathematics Year : 2022

An infinite-dimensional representation of the Ray-Knight theorems

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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 µ-processes, and has an immediate application: a µ-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.
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Dates and versions

hal-03035292 , version 1 (02-12-2020)

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Elie Aïdékon, Yueyun Hu, Zhan Shi. An infinite-dimensional representation of the Ray-Knight theorems. Science China Mathematics, 2022, ⟨10.1007/s11425-022-2068-0⟩. ⟨hal-03035292⟩
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