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Markov processes with identical hitting probabilities. (English) Zbl 0599.60071

Let \((X_ t,P^ x)\) and \((Y_ t,Q^ x)\) be transient right processes (in the sense of Getoor) on a common state space \((E_{\Delta},{\mathcal E}_{\Delta})\), where \(E_{\Delta}=E\cup \{\Delta \}\) and \({\mathcal E}_{\Delta}={\mathcal E}\vee \{\Delta \}\) and \(\Delta\) is a trap for X and Y. For \(B\in {\mathcal E}_{\Delta}\) let \(T_ B\) (resp. \(S_ B)\) be the first hitting time of X (resp. Y) (i.e. \(T_ B=\inf \{t>0\), \(X_ t\in B\})\). Let S(X) (resp. S(Y)) be the cone of excessive functions of X (resp. Y).
The main result of the paper is as follows: Suppose that for each \(B\in {\mathcal E}\), \(P^ x(T_ B<\infty)=Q^ x(S_ B<\infty)\) (x\(\in E)\). Then \(S(X)=S(Y)\), X and Y have identical hitting distributions and each is a time change of the other. This generalizes a result of J. Glover [Trans. Am. Math. Soc. 275, 131-141 (1983; Zbl 0517.60082)].
Reviewer: M.Dozzi

MSC:

60J45 Probabilistic potential theory

Citations:

Zbl 0517.60082
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References:

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