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An infinite-dimensional representation of the Ray-Knight theorems

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Abstract

The classical Ray-Knight theorems for the 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 the Brownian motion. We extend these results by describing the local time process jointly for all a and b, by means of the 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 (2002)), whereas a Feller CSBP with immigration satisfies a stochastic differential equation (SDE) driven by a white noise (Dawson and Li (2012)); our result gives an explicit relation between these two descriptions and shows that the SDE in question is a reformulation of Tanaka’s formula.

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Acknowledgements

This work was supported by ANR MALIN. The authors are grateful to the referees for their careful reading and helpful suggestions which led to improvements in the paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Elie Aïdékon

  2. LAGA, Université Sorbonne Paris Nord, Villetaneuse, 93430, France

    Yueyun Hu

  3. Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Zhan Shi

Authors
  1. Elie Aïdékon
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  2. Yueyun Hu
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  3. Zhan Shi
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Correspondence to Elie Aïdékon.

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Aïdékon, E., Hu, Y. & Shi, Z. An infinite-dimensional representation of the Ray-Knight theorems. Sci. China Math. 67, 149–162 (2024). https://doi.org/10.1007/s11425-022-2068-0

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