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A Ray-Knight theorem for symmetric Markov processes. (English) Zbl 1044.60064
Summary: Let \(X\) be a strongly symmetric recurrent Markov process with state space \(S\) and let \(L^x_t\) 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 \(t>0\), \[ \bigl\{ L^x_{ \tau (t)}+\tfrac 12(\eta_x+b)^2;x\in S\bigr\}=\bigl\{\tfrac 12(\eta_x+\sqrt {2t+b^2})^2;\;x\in S\bigr\}\text{ in law}. \] A version of this theorem is also given when \(X\) is transient.

MSC:
60J55 Local time and additive functionals
60J25 Continuous-time Markov processes on general state spaces
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