Recurrent Dirichlet Forms and Markov Property of Associated Gaussian Fields

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.