Potential kernel for two-dimensional random walk

Abstract

It is proved that the potential kernel of a recurrent, aperiodic random walk on the integer lattice ℤ2 admits an asymptotic expansion of the form (2π√|Q)-1 ln Q(x2, - x1) + const + |x|-1U1(ωx) + |x|-2U2(ωx) + ⋯, where |Q| and Q(θ) are, respectively, the determinant and the quadratic form of the covariance matrix of the increment X of the random walk, ωx = x/|x| and the Uk(ω) are smooth functions of ω, |ω| = 1, provided that all the moments of X are finite. Explicit forms of U1 and U2 are given in terms of the moments of X.