Let $\mathbb{P}:={\left(P_t\right)}_{t>0}$ be a measurable semigroup and $m$ a $\sigma$-finite positive measure on a Lusin space $X$. An $m$-exit law for $\mathbb{P}$ is a family ${\left(f_t\right)}_{t>0}$ of nonnegative measurable functions on $X$ which are finite $m$-a.e. and satisfy for each $s,t>0$ $P_s{f}_t=f_{s+t}$ $m$-a.e. An excessive function $u$ is said to be in $ℛ$ if there exits an $m$-exit law ${\left(f_t\right)}_{t>0}$ for $\mathbb{P}$ such that $u={\int }_0^{\infty }{f}_{t}dt$, $m$-a.e. Let $𝒫$ be the cone of $m$-purely excessive functions with respect to $\mathbb{P}$ and $ℐmV$ be the cone of $m$-potential functions. It is clear that $ℐmV\subseteq ℛ\subseteq 𝒫$. In this...

Let $X$ be a reflexive Banach space and $A$ be a closed operator in an $L_1$-space of $X$-valued functions. Then we characterize the range $R\left(A\right)$ of $A$ as follows. Let $0\ne {\lambda }_n\in \rho \left(A\right)$ for all $1\le n<\infty$, where $\rho \left(A\right)$ denotes the resolvent set of $A$, and assume that ${lim}_{n\to \infty }{\lambda }_n=0$ and ${sup}_{n\ge 1}\parallel {\lambda }_n{({\lambda }_n-A)}^{-1}\parallel <\infty$. Furthermore, assume that there exists ${\lambda }_{\infty }\in \rho \left(A\right)$ such that $\parallel {\lambda }_{\infty }{({\lambda }_{\infty }-A)}^{-1}\parallel \le 1$. Then $f\in R\left(A\right)$ is equivalent to ${sup}_{n\ge 1}{\parallel {({\lambda }_n-A)}^{-1}f\parallel }_1<\infty$. This generalizes Shaw’s result for scalar-valued functions.

A large class of stochastic semilinear equations with measurable nonlinear term on a Hilbert space H is considered. Assuming the corresponding nonsymmetric Ornstein-Uhlenbeck process has an invariant measure μ, we prove in the $L^p(H,\mu )$ spaces the existence of a transition semigroup $\left(P_t\right)$ for the equations. Sufficient conditions are provided for hyperboundedness of $P_t$ and for the Log Sobolev Inequality to hold; and in the case of a bounded nonlinear term, sufficient and necessary conditions are obtained....