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 $t > 0$, $$ \{L^x_{\tau(t)} + \frac{1}{2} (\eta_x + b)^2; x \in S \} = \{ \frac{1}{2} ( \eta_x + \sqrt{2+ b^2})^2 ; x \in S \} \text{ in law}. $$ A version of this theorem is also given when $X$ is transient.