Infinitely divisible processes and their potential theory. II

Sidney C. Port; Charles J. Stone

Annales de l'institut Fourier (1971)

  • Volume: 21, Issue: 4, page 179-265
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Port, Sidney C., and Stone, Charles J.. "Infinitely divisible processes and their potential theory. II." Annales de l'institut Fourier 21.4 (1971): 179-265. <http://eudml.org/doc/74058>.

@article{Port1971,
abstract = {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.},
author = {Port, Sidney C., Stone, Charles J.},
journal = {Annales de l'institut Fourier},
keywords = {probability theory},
language = {eng},
number = {4},
pages = {179-265},
publisher = {Association des Annales de l'Institut Fourier},
title = {Infinitely divisible processes and their potential theory. II},
url = {http://eudml.org/doc/74058},
volume = {21},
year = {1971},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 21
IS - 4
SP - 179
EP - 265
AB - 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.
LA - eng
KW - probability theory
UR - http://eudml.org/doc/74058
ER -

References

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  1. [6] B. BELKIN, Cornell University Thesis, 
  2. [2] R. BLUMENTHAL and R. GETOOR, Markov Processes and Potential Theory, Academic Press, N.Y. (1968). Zbl0169.49204MR41 #9348
  3. [3] N. DUNFORD and J. T. SCHWARTZ, Linear Operators I, Academic Press, N. Y. (1958). Zbl0084.10402MR22 #8302
  4. [4] E. HEWITT and K. A. ROSS, Abstract Harmonic Analysis I, Academic Press, N. Y. (1963). Zbl0115.10603
  5. [5] G. A. HUNT, Markov Processes and Potentials I, Illinois J. Math., 1, 294-319 (1956). 
  6. [6] G. A. HUNT, Markoff Chains and Martin Boundaries, Illinois J. Math. 4, 313-340 (1960). Zbl0094.32103MR23 #A691
  7. [7] S. C. PORT and C. J. STONE, Potential Theory of Random Walks on Abelian Groups. Acta Math., 122, 19-114 (1969). Zbl0183.47201MR41 #6319
  8. [8] S. C. PORT and C. J. STONE, Hitting Times for Transient Random Walks, J. Math. Mech., 17, 1117-1130 (1968). Zbl0162.49201MR37 #2327
  9. [9] S. C. PORT and C. J. STONE, Hitting Times and Hitting Places for non-lattice Recurrent Random Walks, J. Math. Mech., 17, 35-58, (1968). Zbl0187.41202MR35 #6216
  10. [10] C. J. STONE, Ratio Limit Theorems for Random Walks on Groups, Trans. Amer. Math. Soc., 125, 86-100 (1966). Zbl0168.38501MR36 #976
  11. [11] C. J. STONE, Applications of Unsmoothing and Fourier Analysis to Random Walks. Markov Processes and Potential Theory, Ed. by C. Chover. J. Wiley et Sons, N. Y. (1967). Zbl0178.20101
  12. [12] C. J. STONE, On Local and Ratio Limit Theorems. Proceedings of the Fifth Berkeley Symposium on Probability and Statistics. University of California Press, Berkeley, California (1968). Zbl0236.60021
  13. [13] C. J. STONE, Infinite Particle Systems and Multi-Dimensional Renewal Theory, J. Math. Mech., 18, 201-228 (1968). Zbl0165.52603MR37 #7008

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