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Recurrent Dirichlet forms and Markov property of associated Gaussian fields. (English) Zbl 1409.60117

Summary: For the extended Dirichlet space \(\mathcal{F}_e\) of a general irreducible recurrent regular Dirichlet form \((\mathcal{E},\mathcal{F})\) on \(L^2(E;m)\), we consider the family \(\mathbb{G}(\mathcal{E}) = \{X_u;u\in \mathcal{F}_e\}\) of centered Gaussian random variables defined on a probability space \((\Omega, \mathcal{B}, \mathbb{P})\) indexed by the elements of \(\mathcal{F}_e\) and possessing the Dirichlet form \(\mathcal{E}\) as its covariance. We formulate the Markov property of the Gaussian field \(\mathbb{G}(\mathcal{E})\) by associating with each set \(A\subset E\) the sub-\(\sigma\)-field \(\sigma(A)\) of \(\mathcal{B}\) generated by \(X_u\) for every \(u\in \mathcal{F}_e\) whose spectrum \(s(u)\) is contained in \(A\). Under a mild absolute continuity condition on the transition function of the Hunt process associated with \((\mathcal{E}, \mathcal{F})\), we prove the equivalence of the Markov property of \(\mathbb{G}(\mathcal{E})\) and the local property of \((\mathcal{E},\mathcal{F})\). One of the key ingredients in the proof is in that we construct potentials of finite signed measures of zero total mass and show that, for any Borel set \(B\) with \(m(B) > 0\), any function \(u\in \mathcal{F}_e\) with \(s(u) \subset B\) can be approximated by a sequence of potentials of measures supported by \(B\).

MSC:

60J45 Probabilistic potential theory
31C25 Dirichlet forms
60G60 Random fields
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