Recurrent Dirichlet Forms and Markov Property of Associated Gaussian Fields
Tóm tắt
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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