The Annals of Probability

A Ray-Knight theorem for symmetric Markov processes

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 $t > 0$

$$\{L_{\tau(t)} + 1/2 (\eta_x + b)^2; x \in S \} = \{ 1/2 ( \eta_x + \sqrt{2+ b^2})^2 ; x \in S \} \text{ in law}.$$

A version of this theorem is also given when $X$ is transient.

Article information

Source
Ann. Probab., Volume 28, Number 4 (2000), 1781-1796.

Dates
First available in Project Euclid: 18 April 2002

https://projecteuclid.org/euclid.aop/1019160507

Digital Object Identifier
doi:10.1214/aop/1019160507

Mathematical Reviews number (MathSciNet)
MR1813843

Zentralblatt MATH identifier
1044.60064

Subjects
Primary: 60J55: Local time and additive functionals

Citation

Eisenbaum, Nathalie; Kaspi, Haya; Marcus, Michael B.; Rosen, Jay; Shi, Zhan. A Ray-Knight theorem for symmetric Markov processes. Ann. Probab. 28 (2000), no. 4, 1781--1796. doi:10.1214/aop/1019160507. https://projecteuclid.org/euclid.aop/1019160507

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