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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
##### Keywords:
local time; Markov processes; Ray-Knight theorem
Full Text:
##### References:
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