A Ray-Knight theorem for symmetric Markov processes

Let X be a strongly symmetric recurrent Markov process with state space S a nd let L-t(x) denote the local time of X at x is an element of S. For a fix ed element 0 in the state space S, let tau (t) := inf{s: L-s(0) > t}. The 0-potential density, u({o})(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 is an element of 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 classi cal second Ray-Knight theorem: for any b is an element of R and t > 0, {L-r(t)(x) + 1/2(eta (x) + b)(2); x is an element of S} = {1/2(eta (x) + ro ot 2t + b(2))(2); x is an element of S} in law. A version of this theorem is also given when X is transient.