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October 2000 A Ray-Knight theorem for symmetric Markov processes
Nathalie Eisenbaum, Haya Kaspi, Michael B. Marcus, Jay Rosen, Zhan Shi
Ann. Probab. 28(4): 1781-1796 (October 2000). DOI: 10.1214/aop/1019160507

Abstract

Let $X$ be a strongly symmetric recurrent Markov process with state space $S$ and let $L_t^x$ denote the local time of $X$ at $X \in S$. For a fixed element 0 in the state space S, let $$ \tau(t) := \inf \{s: L^0_s > t \}. $$

The 0-potential density, $u_{\{0\}}(x, y)$, of the process $X$ killed at $T_0 = \inf \{s:X_s =0\}$ is symmetric and positive definite. Let $\eta = \{\eta_x; x \in S \}$ be a mean-zero Gaussian process with covariance $$ E_\eta (\eta_x \eta_y ) = u_{\{0\}}(x, y). $$ The main result of this paper is the following generalization of the classical second Ray–Knight theorem: for any $b \in R$ and $t > 0$, $$ \{L^x_{\tau(t)} + \frac{1}{2} (\eta_x + b)^2; x \in S \} = \{ \frac{1}{2} ( \eta_x + \sqrt{2+ b^2})^2 ; x \in S \} \text{ in law}. $$ A version of this theorem is also given when $X$ is transient.

Citation

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Nathalie Eisenbaum. Haya Kaspi. Michael B. Marcus. Jay Rosen. Zhan Shi. "A Ray-Knight theorem for symmetric Markov processes." Ann. Probab. 28 (4) 1781 - 1796, October 2000. https://doi.org/10.1214/aop/1019160507

Information

Published: October 2000
First available in Project Euclid: 18 April 2002

zbMATH: 1044.60064
MathSciNet: MR1813843
Digital Object Identifier: 10.1214/aop/1019160507

Subjects:
Primary: 60J55

Keywords: Local time , Markovprocesses , Ray –Knight theorem

Rights: Copyright © 2000 Institute of Mathematical Statistics

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Vol.28 • No. 4 • October 2000
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