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On a martingale method for symmetric diffusion processes and its applications. (English) Zbl 0717.60090
Let X be a locally compact separable metric space and m be a Radon measure on X whose support is the whole space X. Let (\({\mathcal E},{\mathcal F})\) be a regular symmetric Dirichlet space on \(L^ 2(X,m)\) and denote by M\(=(\Omega,X_ t,P_ x)\) a symmetric Markov process associated with the Dirichlet space (\({\mathcal E},{\mathcal F})\). For \(u\in {\mathcal F}\) denote by \(\tilde u\) the quasi-continuous version of u and let \(A_ t^{[u]}=\tilde u(X_ t)-\tilde u(X_ 0).\)
Under the assumption that the Markov process M is conservative, T. Lyons and W. Zheng [Les processus stochastiques, Coll. Paul Lévy, Palaiseau/Fr. 1987, Astérisque 157-158, 249-271 (1988; Zbl 0654.60059)] obtained another expression of \(A^{[u]}:\) For \(T>0\) \[ (*)\quad A_ t^{[u]}=M_ t^{[u]}-(M_ T^{[u]}(r_ T)- M^{[u]}_{T-t}(r_ T)),\quad 0\leq t\leq T,\quad P_ m-a.e. \] where \(r_ T\) is a time reverse operator at T, i.e., \(X_ t(r_ T)=X_{T- t}\), and \(P_ m\) is a \(\sigma\)-finite measure defined by \(\int_{X}P_ x[\cdot]dm.\)
One can say that the decomposition (*) reflects the symmetry of the Markov process M faithfully. Furthermore (*) would enable us to use the martingale theory in the study of symmetric Markov processes. The purpose of the present paper is to demonstrate this in getting a conservativeness criterion, a tightness criterion and also some sample path properties for symmetric diffusion processes. We shall further consider an extension of the method to non-symmetric situations.

60J60 Diffusion processes
60G44 Martingales with continuous parameter