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.
@article{119286,
author = {Imed Bachar},
title = {On exit laws for semigroups in weak duality},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {42},
year = {2001},
pages = {711-719},
zbl = {1090.31501},
mrnumber = {1883379},
language = {en},
url = {http://dml.mathdoc.fr/item/119286}
}
Bachar, Imed. On exit laws for semigroups in weak duality. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) pp. 711-719. http://gdmltest.u-ga.fr/item/119286/
Potential theory on locally compact Abellian Groups, Springer-Verlag, Berlin-Heidelberg-New York, 1975. | MR 0481057
Order and Convexity in Potential Theory, Lecture Notes in Math. 853, Springer, Berlin-Heidelberg-New York, 1980.
Probabilités et Potentiel, Chapter $XII-XVI$, Herman, 1987. | MR 0488194 | Zbl 0624.60084
Markov processes and non symmetric Dirichlet forms without regularity, J. Funct. Anal. 85 287-306 (1989). (1989) | MR 1012207
On the potential theory of symmetric Markov processes, Math. Ann. 281 495-512 (1988). (1988) | MR 0954155 | Zbl 0627.60067
Dirichlet Forms and Markov Processes, North-Holland, Amsterdam-Oxford-New York, 1980. | MR 0569058 | Zbl 0422.31007
Excessive Measures, Birkhäuser Processes, 1990. | MR 1093669 | Zbl 1081.60544
Riesz decomposition in Markov process theory, Trans. Amer. Math. Soc. 285 107-132 (1989). (1989) | MR 0748833
Naturality standardness and weak duality for Markov processes, Z. Wahrsch verw. Gebiete 67 1-62 (1984). (1984) | MR 0756804 | Zbl 0553.60070
Lois de sortie et semi-groupes basiques, Manuscripta Math. 75 293-302 (1992). (1992) | MR 1167135 | Zbl 0759.60080
Sur la représentation par les lois de sortie, Math. Z. 213 647-656 (1993). (1993) | MR 1231882 | Zbl 0790.31006
On the functional equation of exit laws for lattice semi-groups, Ann. Ecole Normale Superieure de Cracowie 196 63-72 (1998). (1998) | MR 1826075
Representation of excessive measures, Sem. Stoch. Processes Birkhäuser, Boston, Mass., 1987, pp.85-105. | MR 0902428 | Zbl 0619.47035
Symmetric Markov Processes, Lecture Notes in Math. 426, Springer, Berlin-Heidelberg-New York, 1974. | MR 0386032 | Zbl 0331.60046