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semigroup; weak duality; exit law
Let $\Bbb P:=(P_{t})_{t>0}$ be a measurable semigroup and $m$ a $\sigma $-finite positive measure on a Lusin space $X$. An $m$-exit law for $\Bbb P$ is a family $(f_{t})_{t>0}$ 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 $\Cal R$ if there exits an $m$-exit law $(f_{t})_{t>0}$ for $\Bbb P$ such that $u=\int_{0}^{\infty }f_{t}\,dt$, $m$-a.e. Let $\Cal P$ be the cone of $m$-purely excessive functions with respect to $\Bbb P$ and $\Cal I mV$ be the cone of $m$-potential functions. It is clear that $\Cal I mV\subseteq \Cal R\subseteq \Cal P$. In this paper we are interested in the converse inclusion. We extend some results already obtained under the assumption of the existence of a reference measure. Also, we give an integral representation of the mutual energy function.
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