On the generalized Feynman-Kac transformation for nearly symmetric Markov processes

Abstract

Suppose $X$ 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: $\tilde{u}(X_t)-\tilde{u}(X_0)=M_t^u+N_t^u$. In this paper, we investigate the strong continuity of the generalized Feynman-Kac semigroup defined by $P_t^{u}f(x)=E_x[e^{N_t^u}f(X_t)]$. Let $Q^u(f,g)=\mathcal{E}(f,g)+\mathcal{E}(u,fg)$ for $f,g\in D(\mathcal{E}{)}_b$. Denote by $J_1$ 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 ${\alpha }_0\ge 0$ such that $\Vert P_t^u{\Vert }_2\le e^{{\alpha }_{0}t}$ for every $t>0$. If one of these conditions holds, then $(P_t^u{)}_{t\ge 0}$ 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.