Cette deuxième partie de notre travail sur les processus i.d. a quatre buts principaux :
(1) Construire un opérateur potentiel pour les processus i.d. (indéfiniment divisible) récurrents et utiliser cet opérateur pour trouver le comportement asymptotique des lois d’atteinte et des fonctions de Green pour des ensembles relativement compacts.
(2) Développer la notion appropriée de mesure d’équilibre et de la constante de Robin pour les ensembles boréliens ;
(3) Établir le comportement asymptotique dans le temps de la loi jointe du temps d’atteinte et de la position d’atteinte d’ensembles relativement compacts.
(4) Examiner diverses questions de théorie du potentiel à la fois pour des processus transitoires et pour des processus récurrents. Nous trouvons notamment toutes les solutions d’équations du type de Poisson qui sont bornées inférieurement lorsque le rôle du laplacien est joué par le générateur infinitésimal du processus ou du processus arrêté sur un ensemble fermé. Nous trouvons aussi toutes les solutions bornées inférieurement pour des problèmes de Dirichlet sur des ensembles fermés.
This second part of our two part work on i.d. process has four main goals:
(1) To develop a potential operator for recurrent i.d. (infinitely divisible) processes and to use this operator to find the asymptotic behavior of the hitting distribution and Green’s function for relatively compact sets in the recurrent case.
(2) To develop the appropriate notion of an equilibrium measure and Robin’s constant for Borel sets.
(3) To establish the asymptotic behavior questions of a potential theoretic nature for both transient and recurrent processes. These include finding all solutions of Poisson type equations that are bounded from below, where the role of the Laplace operator is played by either the infinitesimal generator of the process or of the process stopped on a closed set, and of finding all solutions that are bounded from below for Dirichlet type problems on closed sets.
@article{AIF_1971__21_4_179_0,
author = {Port, Sidney C. and Stone, Charles J.},
title = {Infinitely divisible processes and their potential theory. {II}},
journal = {Annales de l'Institut Fourier},
pages = {179--265},
publisher = {Institut Fourier},
address = {Grenoble},
volume = {21},
number = {4},
year = {1971},
doi = {10.5802/aif.398},
mrnumber = {346919},
zbl = {0215.53401},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/aif.398/}
}
TY - JOUR AU - Port, Sidney C. AU - Stone, Charles J. TI - Infinitely divisible processes and their potential theory. II JO - Annales de l'Institut Fourier PY - 1971 SP - 179 EP - 265 VL - 21 IS - 4 PB - Institut Fourier PP - Grenoble UR - http://archive.numdam.org/articles/10.5802/aif.398/ DO - 10.5802/aif.398 LA - en ID - AIF_1971__21_4_179_0 ER -
%0 Journal Article %A Port, Sidney C. %A Stone, Charles J. %T Infinitely divisible processes and their potential theory. II %J Annales de l'Institut Fourier %D 1971 %P 179-265 %V 21 %N 4 %I Institut Fourier %C Grenoble %U http://archive.numdam.org/articles/10.5802/aif.398/ %R 10.5802/aif.398 %G en %F AIF_1971__21_4_179_0
Port, Sidney C.; Stone, Charles J. Infinitely divisible processes and their potential theory. II. Annales de l'Institut Fourier, Tome 21 (1971) no. 4, pp. 179-265. doi : 10.5802/aif.398. http://archive.numdam.org/articles/10.5802/aif.398/
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