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Wiener's test for space-time random walks and its applications
Author(s):
Yasunari
Fukai;
Kôhei
Uchiyama
Journal:
Trans. Amer. Math. Soc.
348
(1996),
4131-4152.
MSC (1991):
Primary 60J15, 60J45, 31C20
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Abstract:
This paper establishes a criterion for whether a  -dimensional random walk on the integer lattice  visits a space-time subset infinitely often or not. It is a precise analogue of Wiener's test for regularity of a boundary point with respect to the classical Dirichlet problem. The test obtained is applied to strengthen the harder half of Kolmogorov's test for the random walk.
References:
- [E]
- Erdös, P., On the law of the iterated logarithm, Ann. of Math. 43 (1942), 419-436. MR 4:16j
- [EG]
- Evans, L.C., Gariepy, R.F., Wiener's criterion for the heat equation, Arch. Rat. Mech. Anal. 78 (1982), 293-314. MR 83g:35047
- [F1]
- Feller, W., The general form of the so-called law of the iterated logarithm, Trans. Amer. Math. Soc. 54 (1943), 373-402. MR 5:125c
- [F2]
- Feller, W., The law of the iterated logarithm for identically distributed random variables, Ann. Math. 47 (1946), 631-638. MR 8:214h
- [F3]
- Feller, W., An introduction to probability theory and its applications, Vol. I, 2nd ed., John Wiley and Sons (1957). MR 19:466a
- [GK]
- Gnedenko, B.V., Kolmogorov, A.N., Limit distributions for sums of independent random variables (translated from Russian), Addison-Wesley, Reading, MA, 1954. MR 16:52d
- [GL]
- Garofalo, N., Lanconelli, E., Wiener's criterion for parabolic equations with variable coefficients and its consequences, Trans. Amer. Math. Soc. 308 (1988), 811-836. MR 89k:35104
- [IM]
- Ito, K., McKean, H.P., Potentials and the random walk, Illinois J. Math. 4 (1960), 119-132. MR 22:12317
- [Ld]
- Landis, E.M., Necessary and sufficient conditions for regularity of a boundary point in the Dirichlet problem for the heat-conduction equation, Soviet Math. 10 (1969), 380-384. MR 41:7308 (of Russian original)
- [Lm]
- Lamperti, J., Wiener's test and Markov chains, J. Math. Anal. Appl. 6 (1963), 58-66. MR 26:817
- [Lv]
- Lévy, P., Théorie de l'addition des variables aléatoires, Paris: Gautier-Villars (1937).
- [M]
- Motoo, M, Proof of the law of iterated logarithm through diffusion equation, Ann. Inst. Statis. Math. 10 (1959), 21-28. MR 20:4331
- [P]
- Petrovskii, I., Zur ersten Randwertaufgabe der Wärmeleitungsgleichung, Compos. Math. 1 (1935), 383-419.
- [S]
- Spitzer, F., Principles of random walk, 2nd ed., Springer-Verlag (1976). MR 52:9383
- [U]
- Uchiyama, K., A probabilistic proof and applications of Wiener's test for the heat operator, Math. Ann. 283 (1989), 65-86. MR 90b:60099
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Additional Information:
Yasunari
Fukai
Affiliation:
Department of Applied Physics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152, Japan
Email:
uchiyama@neptune.ap.titech.ac.jp
Kôhei
Uchiyama
Affiliation:
Department of Applied Physics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152, Japan
DOI:
10.1090/S0002-9947-96-01643-1
PII:
S 0002-9947(96)01643-1
Keywords:
Wiener's test,
random walk,
Kolmogorov's test,
discrete heat equation,
regularity of a minimal Martin boundary point
Received by editor(s):
May 10, 1995
Copyright of article:
Copyright
1996,
American Mathematical Society
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