# Infinitely divisible processes and their potential theory. I

Sidney C. Port; Charles J. Stone

Annales de l'institut Fourier (1971)

- Volume: 21, Issue: 2, page 157-275
- ISSN: 0373-0956

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topPort, Sidney C., and Stone, Charles J.. "Infinitely divisible processes and their potential theory. I." Annales de l'institut Fourier 21.2 (1971): 157-275. <http://eudml.org/doc/74032>.

@article{Port1971,

abstract = {We show that associated with every i.d. (infinitely divisible) process on a locally compact, non-compact 2nd countable Abelian group is a corresponding potential theory that yields definitive results on the behavior of the process in both space and time. Our results are general, no density or other smoothness assumptions are made on the process. In this first part of two part work we have four main goals.(1) To lay the probabilistic foundation of such processes. This mainly consists in giving the classification into transient and recurrent processes, examining the time periodicities of the process and developing appropriate ratio limit theorems for the transition function, and establishing the revewal theorem for transient processes.(2) To develop and apply a $\lambda $ capacity theory that gives necessary and sufficient conditions for a Borel set to be essentially polar.(3) To investigate the asymptotic behavior of the hitting distribution and Green’s function for Borel sets in the transient case. To do this it is first necessary to develop the notion of transient and recurrent sets and show that each transient set has a unique capacitory measure associated with it.(4) To investigate the asymptotic behavior for large time of the joint distribution of the first (last) hitting time and first (last) hitting place for a Borel set in the transient case.},

author = {Port, Sidney C., Stone, Charles J.},

journal = {Annales de l'institut Fourier},

keywords = {probability theory},

language = {eng},

number = {2},

pages = {157-275},

publisher = {Association des Annales de l'Institut Fourier},

title = {Infinitely divisible processes and their potential theory. I},

url = {http://eudml.org/doc/74032},

volume = {21},

year = {1971},

}

TY - JOUR

AU - Port, Sidney C.

AU - Stone, Charles J.

TI - Infinitely divisible processes and their potential theory. I

JO - Annales de l'institut Fourier

PY - 1971

PB - Association des Annales de l'Institut Fourier

VL - 21

IS - 2

SP - 157

EP - 275

AB - We show that associated with every i.d. (infinitely divisible) process on a locally compact, non-compact 2nd countable Abelian group is a corresponding potential theory that yields definitive results on the behavior of the process in both space and time. Our results are general, no density or other smoothness assumptions are made on the process. In this first part of two part work we have four main goals.(1) To lay the probabilistic foundation of such processes. This mainly consists in giving the classification into transient and recurrent processes, examining the time periodicities of the process and developing appropriate ratio limit theorems for the transition function, and establishing the revewal theorem for transient processes.(2) To develop and apply a $\lambda $ capacity theory that gives necessary and sufficient conditions for a Borel set to be essentially polar.(3) To investigate the asymptotic behavior of the hitting distribution and Green’s function for Borel sets in the transient case. To do this it is first necessary to develop the notion of transient and recurrent sets and show that each transient set has a unique capacitory measure associated with it.(4) To investigate the asymptotic behavior for large time of the joint distribution of the first (last) hitting time and first (last) hitting place for a Borel set in the transient case.

LA - eng

KW - probability theory

UR - http://eudml.org/doc/74032

ER -

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