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

Fukai, Yasunari; Uchiyama, Kôhei

doi:10.1090/S0002-9947-96-01643-1

Wiener’s test for space-time random walks and its applications
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by Yasunari Fukai and Kôhei Uchiyama

Trans. Amer. Math. Soc. 348 (1996), 4131-4152

DOI: https://doi.org/10.1090/S0002-9947-96-01643-1

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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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