Abstract
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 ⊂ E the sub-σ-field σ(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) ⊂ B can be approximated by a sequence of potentials of measures supported by B.
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Fukushima, M., Oshima, Y. Recurrent Dirichlet Forms and Markov Property of Associated Gaussian Fields. Potential Anal 49, 609–633 (2018). https://doi.org/10.1007/s11118-017-9669-z
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DOI: https://doi.org/10.1007/s11118-017-9669-z