Local property of Dirichlet forms and diffusions on general state spaces (Q1318088)

The well-known result of Fukushima, that a Hunt process on a locally compact separable metric space associated with a regular Dirichlet form \(({\mathfrak E},D({\mathfrak E}))\) is a diffusion if and only if the Dirichlet form \(({\mathfrak E},D({\mathfrak E}))\) has the local property, is extended to more general Markov processes and (nonsymmetric) Dirichlet forms. Let \(M\) be a special standard process with lifetime \(\zeta\) and a state space \(E\) being a (metrizable) Lusin space. Theorem. If \(M\) is associated with \(({\mathfrak E},D({\mathfrak E}))\), a coercive closed form on \(L^ 2(E,m)\) (\(m\) being an appropriate measure) that possesses the Dirichlet property, then \(({\mathfrak E},D({\mathfrak E}))\) has the local property if and only if \[ P_ z(t\to X_ t\text{ is continuous on }[0,\zeta))=1 \text{ for }{\mathfrak E}\text{-q.e. } z\in E. \] In particular, \(M\) can be modified outside some \(M\)-invariant set with \({\mathfrak E}\)-exceptional complement, so that \(M\) becomes a diffusion.