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Green's functions for random walks onℤN

Published online by Cambridge University Press:  01 July 1998

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Abstract

Green's function $G(x)$ of a zero mean random walk on the $N$-dimensional integer lattice ($N\geq 2$) is expanded in powers of $1/|x|$ under suitable moment conditions. In particular, we find minimal moment conditions for $G(x)$ to behave like a constant times the Newtonian potential (or logarithmic potential in two dimensions) for large values of $|x|$. Asymptotic estimates of $G(x)$ in dimensions $N\geq 4$, which are valid even when these moment conditions are violated, are computed. Such estimates are applied to determine the Martin boundary of the random walk. If $N= 3$ or $4$ and the random walk has zero mean and finite second moment,the Martin boundary consists of one point, whereas if $N\geq 5$, this is not the case, because non-harmonic functions arise as Martin boundary points for a large class of such random walks. A criterion for when this happens is provided.

1991 Mathematics Subject Classification: 60J15, 60J45, 31C20.

Type
Research Article
Copyright
London Mathematical Society 1998

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