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On the generalized Feynman-Kac transformation for nearly symmetric Markov processes. (English) Zbl 1252.31008

Summary: Suppose that \(X\) is a right process which is associated with a non-symmetric Dirichlet form \((\mathcal{E},D(\mathcal{E}))\) on \(L^2(E;m)\). For \(u\in D(\mathcal{E})\), we have Fukushima’s decomposition: \(\tilde{u}(X_{t})-\tilde{u}(X_{0})=M^{u}_{t}+N^{u}_{t}\). In this paper, we investigate the strong continuity of the generalized Feynman-Kac semigroup defined by \(P^{u}_{t}f(x)=E_{x}\big[e^{N^{u}_{t}}f(X_{t})\big]\). Let \(Q^{u}(f,g)=\mathcal{E}(f,g)+\mathcal{E}(u,fg)\) for \(f,g\in D(\mathcal{E})_{b}\). Denote by \(J_1\) the dissymmetric part of the jumping measure \(J\) of \((\mathcal{E},D(\mathcal{E}))\). Under the assumption that \(J_1\) is finite, we show that \((Q^{u},D(\mathcal{E})_{b})\) is lower semi-bounded if and only if there exists a constant \(\alpha_0\geq 0\) such that \(\|P^{u}_{t}\|_{2}\leq e^{\alpha_{0}t}\) for every \(t>0\). If one of these conditions holds, then \((P^{u}_{t})_{t\geq0}\) is strongly continuous on \(L^2(E;m)\). If \(X\) is equipped with a differential structure, then this result also holds without assuming that \(J_1\) is finite.

MSC:

31C25 Dirichlet forms
60J57 Multiplicative functionals and Markov processes
60J45 Probabilistic potential theory
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