A Ray-Knight theorem for symmetric Markov processes

Abstract

Let X be a strongly symmetric recurrent Markov process with state space S and let Lxt denote the local time of X at x ∈ S. For a fixed element 0 in the state space S, let τ(t) := inf {s: L0s > t}. The 0-potential density, u{0}(x, y), of the process X killed at T0 = inf{s: Xs = 0}, is symmetric and positive definite. Let η = {ηx; x ∈ S} be a mean-zero Gaussian process with covariance Eη(ηxηy) = u{0}(x, y). The main result of this paper is the following generalization of the classical second Ray-Knight theorem: for any b ∈ R and t > 0, {Lxτ(t) + 1/2(ηx + b)2; x ∈ S} = {1/2(ηx + √2t + b2)2; x ∈ S} in law. A version of this theorem is also given when X is transient.