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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 $$X^0$$.

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