# 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

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topPort, 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

top- [6] B. BELKIN, Cornell University Thesis,
- [2] R. BLUMENTHAL and R. GETOOR, Markov Processes and Potential Theory, Academic Press, N.Y. (1968). Zbl0169.49204MR41 #9348
- [3] N. DUNFORD and J. T. SCHWARTZ, Linear Operators I, Academic Press, N. Y. (1958). Zbl0084.10402MR22 #8302
- [4] E. HEWITT and K. A. ROSS, Abstract Harmonic Analysis I, Academic Press, N. Y. (1963). Zbl0115.10603
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- [6] G. A. HUNT, Markoff Chains and Martin Boundaries, Illinois J. Math. 4, 313-340 (1960). Zbl0094.32103MR23 #A691
- [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] S. C. PORT and C. J. STONE, Hitting Times for Transient Random Walks, J. Math. Mech., 17, 1117-1130 (1968). Zbl0162.49201MR37 #2327
- [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] C. J. STONE, Ratio Limit Theorems for Random Walks on Groups, Trans. Amer. Math. Soc., 125, 86-100 (1966). Zbl0168.38501MR36 #976
- [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] 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] 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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