Quasi-regular Dirichlet forms and Markov processes (Q1803342)

A fundamental result of \textit{M. Fukushima} [Dirichlet forms and Markov processes (1880; Zbl 0422.31007)] says that there exists uniquely (up to q.e. equivalence) a Hunt process associated with a regular Dirichlet form \(({\mathcal E},{\mathcal F})\) on a locally compact space \(X\). But the converse assertion is not true in general. The main result of this paper is to give an analytic characterization of the (not necessarily symmetric) Dirichlet form on a metrizable Lusin state space \(X\) which is associated with a dual pair of special standard (or right) processes \((X,\widehat X)\). More precisely, \(({\mathcal E},{\mathcal F})\) is called quasi-regular if there exists an \({\mathcal E}\)-nest of compact subsets and an \(\widetilde {\mathcal E}_ 1^{1/2}\)-dense subset of \(D({\mathcal E})\) whose elements have \({\mathcal E}\)-q.c. versions, where \(\widetilde{\mathcal E} (u,v)={1\over 2}({\mathcal E}(u,v)+{\mathcal E}(v,u))\). Then a necessary and sufficient condition on \(({\mathcal E},{\mathcal F})\) for the association of the special standard pair \((X,\widehat X)\) is the quasi-regularity. There is a work of \textit{Y. Le Jan} [J. Math. Soc. Jap. 35, 37-42 (1983; Zbl 0528.60070)] in this connection.