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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Wiener's test for space-time random walks
and its applications


Authors: Yasunari Fukai and 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 $d$-dimensional random walk on the integer lattice $ \mathbf {Z}^{d}$ 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.


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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: http://dx.doi.org/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
Article copyright: © Copyright 1996 American Mathematical Society