On exit laws for semigroups in weak duality
Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) no. 4, pp. 711-719
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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.
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.
Bachar, Imed. On exit laws for semigroups in weak duality. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) no. 4, pp. 711-719. http://geodesic.mathdoc.fr/item/CMUC_2001_42_4_a10/
@article{CMUC_2001_42_4_a10,
author = {Bachar, Imed},
title = {On exit laws for semigroups in weak duality},
journal = {Commentationes Mathematicae Universitatis Carolinae},
pages = {711--719},
year = {2001},
volume = {42},
number = {4},
mrnumber = {1883379},
zbl = {1090.31501},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMUC_2001_42_4_a10/}
}