A Characterization of the Kernel $\lim{_\lambda \downarrow 0}V_\lambda$ for Sub-Markovian Resolvents $(V_\lambda)^1$

Abstract

Let $(V_\lambda)$ be a sub-Markovian resolvent of kernels $V_\lambda$ on a measurable space $(E, \mathscr{E})$. Assume that $V = \lim_{\lambda \downarrow 0}V_\lambda$ is a proper kernel. The proper kernels $V$ on $(E, \mathscr{E})$ that are of the form $V = \lim_{\lambda \downarrow 0}V_\lambda, (V_\lambda)$ a sub-Markovian resolvent of kernels on $(E, \mathscr{E})$, are proved to be precisely those proper kernels $V$ which satisfy the complete maximum principle and for which the following condition holds: there exists an increasing sequence $(A_n) \subset \mathscr{E}$ with $\mathbf{\bigcup}_n A_n = E$ such that (i) $V1_{A_n} < \infty$ for all $n$; and (ii) if $f \in \mathscr{E}^+$ and $Vf < \infty$ then $\inf_nR_{\mathscr{C} A_n} Vf < \infty$, where $R_Bu = \inf \{v \text{supermedian} \mid u \geqq v \text{on} B\}$.