The Annals of Probability

Potential kernel for two-dimensional random walk

Yasunari Fukai and Kôhei Uchiyama

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 6 January 2003

Permanent link to this document
http://projecteuclid.org/euclid.aop/1041903213

Digital Object Identifier
doi:10.1214/aop/1041903213

Mathematical Reviews number (MathSciNet)
MR1415236

Zentralblatt MATH identifier
0879.60068

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

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

Citation

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


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References

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