On the Generalized Feynman–Kac Transformation for Nearly Symmetric Markov Processes

Abstract

Suppose that X is a right process which is associated with a non-symmetric Dirichlet form \((\mathcal{E},D(\mathcal{E}))\) on L2(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 \(P^{u}_{t}f(x)=E_{x}[e^{N^{u}_{t}}f(X_{t})]\). Let \(Q^{u}(f,g)=\mathcal{E}(f,g)+\mathcal{E}(u,fg)\) for \(f,g\in D(\mathcal{E})_{b}\). Denote by J1 the dissymmetric part of the jumping measure J of \((\mathcal{E},D(\mathcal{E}))\). Under the assumption that J1 is finite, we show that \((Q^{u},D(\mathcal{E})_{b})\) is lower semi-bounded if and only if there exists a constant α0≥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 L2(E;m). If X is equipped with a differential structure, then this result also holds without assuming that J1 is finite.