Girsanov and Feynman-Kac type transformations for symmetric Markov processes
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 \({\mathbf M}=(X_t,P_x)\) and its associated regular Dirichlet form \(({\mathcal E},{\mathcal F})\) on \(L^2(E;m)\). For a quasi-continuous version \(\widetilde{u}\) of \(u \in {\mathcal F}_e\), 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 \((Q,{\mathcal D}(Q))\) 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, \({\mathcal D}(Q)={\mathcal F}\) and \(Q(f,g)={\mathcal E}(f,g) + {\mathcal E}(u,fg)\) for any \(f,g \in {\mathcal F}_b\). 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 \(e^{A_t}\).
scientific article; zbMATH DE number 1783423
scientific article