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Poisson point processes attached to symmetric diffusions. (English) Zbl 1083.60065
Let \(a\) be a non-isolated point of a topological space \(S\) and \(X^0=(X^0_t, 0 \leq t < \zeta^0, P^0_x)\) be a symmetric diffusion on \(S_0 = S \setminus \{ a \}\) such that \(P^0_x(\zeta^0 < \infty, X^0_{\zeta^0-}=a)>0\) for \(x \in S_0\). By making use of Poisson point processes taking values in the spaces of excursions around \(a\) whose characteristic measures are uniquely determined by \(X^0\), a symmetric diffusion \({\widetilde X}\) on \(S\) with no killing inside \(S\) which extends \(X^0\) on \(S_0\) is constructed. It is also proved that such a process \({\widetilde X}\) is unique in law and its resolvent and Dirichlet form admit explicit expressions in terms of \(X^0\).

60J60 Diffusion processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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