Recurrence and conservativeness of symmetric diffusion processes by Girsanov transformations
Let \(X\) be a separable metric space and \(m\) a \(\sigma\)-finite Borel measure on \(X\). Let \(({\mathcal E},{\mathcal F})\) be a quasi-regular Dirichlet space on \(L^2(X;m)\) with associated diffusion process \({\mathbb M}=(X_t,P_x)\) having no killing inside. For a nonnegative quasi-continuous function \(\varphi\) belonging to \({\mathcal F}\) locally in the broad sense, let \(\tau\) be the exit time from \(X_\varphi=\{x \in X:0<\varphi(x)<\infty\}\). Denote by \({\mathbb M}^\varphi=(X_t,P^\varphi_x)\) the process on \(X_\varphi\) transformed by the multiplicative functional \(L^{[\varphi]}_t= \exp[M^{[\log \varphi]}_t-{1 \over 2} \langle M^{[\log \varphi]}\rangle_t] I_{\{t \leq \tau\}}\). The main purpose of this paper is to give the recurrence and conservativeness criteria of \({\mathbb M}^\varphi\) by using the growth of the volume of the ball determined by the intrinsic metric. The criteria given in this paper are generalizations of those given in the \(C_0\)-regular case by \textit{K.-Th. Sturm} [J. Reine Angew. Math. 456, 173-196 (1994; Zbl 0806.53041)] and \textit{H. Ôkura} [in: Dirichlet forms and stochastic processes, 291-303 (1995; Zbl 0840.60071)]. Some examples which cannot be covered by the framework of the regular Dirichlet spaces are also given.
scientific article; zbMATH DE number 1295315