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Zbl 1004.60077
Chen, Zhen-Qing; Zhang, Tu-Sheng
Girsanov and Feynman-Kac type transformations for symmetric Markov processes. (English)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 38, No.4, 475-505 (2002). ISSN 0246-0203

Let $E$ be a locally compact separable metric space and $m$ a positive Radon measure on $E$ with full support. Consider an $m$-symmetric Markov process ${\bold M}=(X_t,P_x)$ and its associated regular Dirichlet form $({\cal E},{\cal F})$ on $L^2(E;m)$. For a quasi-continuous version $\widetilde{u}$ of $u \in {\cal F}_e$, let $\widetilde{u}(X_t)-\widetilde{u}(X_0)= M^u_t+N^u_t$ be the Fukushima decomposition into a martingale additive functional $M^u_t$ and a continuous additive functional $N^u_t$ of zero energy. Let $\widehat{P}_t$ be the Feynman-Kac type transition function defined by $\widehat{P}_t f(x)=E_x\left(f(X_t)e^{N^u_t}\right)$. Note that $N^u_t$ is not of finite variation in general. The main result of this paper is to characterize the quadratic form $(Q,{\cal D}(Q))$ associated with $\widehat{P}_t$. The result says that, if the Revuz measure $\mu_{\langle u\rangle}$ of $\langle M^u\rangle$ is in the Kato class, then $\widehat{P}_t$ is $m$-symmetric, ${\cal D}(Q)={\cal F}$ and $Q(f,g)={\cal E}(f,g) + {\cal E}(u,fg)$ for any $f,g \in {\cal F}_b$. Since $e^{N^u_t}=e^{\widetilde{u}(X_t)-\widetilde{u}(X_0)} e^{A_t}L_t$ for a jump type exponential martingale $L_t$ and a continuous additive functional $A_t$ of finite variation, the proof is performed by characterizing the transformed process by $L_t$ first and then using the Feynman-Kac transformation by $e^{A_t}$.
[Yoichi Oshima (Kumamoto)]

MSC 2000:: *60J45 Probabilistic potential theory
60J57 Multiplicative functionals
31C25 Dirichlet spaces

Keywords: Dirichlet forms; exponential martingales; Girsanov transformations

Cited in: Zbl 1118.60067

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