On exit laws for semigroups in weak duality

@article{Bachar2001,
abstract = {Let $\mathbb \{P\}:=(P_\{t\})_\{t>0\}$ be a measurable semigroup and $m$ a $\sigma $-finite positive measure on a Lusin space $X$. An $m$-exit law for $\mathbb \{P\}$ 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 $\mathcal \{R\}$ if there exits an $m$-exit law $(f_\{t\})_\{t>0\}$ for $\mathbb \{P\}$ such that $u=\int _\{0\}^\{\infty \}f_\{t\}\,dt$, $m$-a.e. Let $\mathcal \{P\}$ be the cone of $m$-purely excessive functions with respect to $\mathbb \{P\}$ and $\mathcal \{I\} mV$ be the cone of $m$-potential functions. It is clear that $\mathcal \{I\} mV\subseteq \mathcal \{R\}\subseteq \mathcal \{P\}$. 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.},
author = {Bachar, Imed},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {semigroup; weak duality; exit law; semigroup; weak duality; exit law},
language = {eng},
number = {4},
pages = {711-719},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On exit laws for semigroups in weak duality},
url = {http://eudml.org/doc/248777},
volume = {42},
year = {2001},
}

TY - JOUR
AU - Bachar, Imed
TI - On exit laws for semigroups in weak duality
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2001
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 42
IS - 4
SP - 711
EP - 719
AB - Let $\mathbb {P}:=(P_{t})_{t>0}$ be a measurable semigroup and $m$ a $\sigma $-finite positive measure on a Lusin space $X$. An $m$-exit law for $\mathbb {P}$ 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 $\mathcal {R}$ if there exits an $m$-exit law $(f_{t})_{t>0}$ for $\mathbb {P}$ such that $u=\int _{0}^{\infty }f_{t}\,dt$, $m$-a.e. Let $\mathcal {P}$ be the cone of $m$-purely excessive functions with respect to $\mathbb {P}$ and $\mathcal {I} mV$ be the cone of $m$-potential functions. It is clear that $\mathcal {I} mV\subseteq \mathcal {R}\subseteq \mathcal {P}$. 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.
LA - eng
KW - semigroup; weak duality; exit law; semigroup; weak duality; exit law
UR - http://eudml.org/doc/248777
ER -