The Annals of Probability

Potential kernel for two-dimensional random walk

Yasunari Fukai and Kôhei Uchiyama

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It is proved that the potential kernel of a recurrent, aperiodic random walk on the integer lattice $\mathbb{Z}^2$ admits an asymptotic expansion of the form $$(2 \pi \sqrt{|Q|})^{-1} \ln Q(x_2, -x_1) + \const + |x|^{-1} U_1 (\omega^x) + |x|^{-2} U_2 (\omega^x) + \dots ,$$ where $|Q|$ and $Q(\theta)$ are, respectively, the determinant and the quadratic form of the covariance matrix of the increment X of the random walk, $\omega^x = x/|x|$ and the $U_k (\omega)$ are smooth functions of $\omega, |\omega| = 1$, provided k that all the moments of X are finite. Explicit forms of $U_1$ and $U_2$ are given in terms of the moments of X.

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Ann. Probab. Volume 24, Number 4 (1996), 1979-1992.

First available in Project Euclid: 6 January 2003

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Zentralblatt MATH identifier

Primary: 60J15 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 31C20: Discrete potential theory and numerical methods

Two-dimensional random walk potential kernel Laplace discrete operator oscillatory integral


Fukai, Yasunari; Uchiyama, Kôhei. Potential kernel for two-dimensional random walk. Ann. Probab. 24 (1996), no. 4, 1979--1992. doi:10.1214/aop/1041903213.

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  • 1 KESTEN, H. 1987. Hitting probabilities of random walks on Z. Stochastic Process. Appl. 25 165 184.
  • 2 LAWLER, G. F. 1991. Intersections of Random Walks. Birkhauser, Boston. ¨
  • 3 SPITZER, F. 1976. Principles of Random Walk, 2nd ed. Springer, New York. ¨
  • 4 STOHR, A. 1950. Uber einige lineare partielle Differenzengleichungen mit konstanten Ko¨ effizienten. III. Math. Nachr. 3 330 357.