The difference between a discrete and continuous harmonic measure

Jianping Jiang, Tom Kennedy

Published 2015-06-13Version 1

Let $D$ be a simply connected domain in the plane. Consider a nearest neighbor random walk on a lattice with spacing $h$. It is well-known that the distribution of the point where the walk exits the domain converges weakly to harmonic measure on $\partial D$ as $h\downarrow 0$. A natural question is to find the leading order term in the difference between harmonic measure and the exit distribution for the random walk. In this paper we answer this question for a random walk that moves in the continuum with steps which are uniformly distributed over a disk of radius $h$. For domains $D$ with analytic boundary, we prove there is a bounded continuous function $\sigma_D(z)$ on the boundary of $D$ such that for functions $g$ which are in $C^{2+\alpha}(\partial D)$ for some $\alpha>0$ $$ \lim_{h\downarrow 0} \frac{\int_{\partial D} g(\xi) \omega_h(0,|d\xi|;D) -\int_{\partial D} g(\xi)\omega(0,|d\xi|;D)}{h} = \int_{\partial D}g(z) \sigma_D(z) |dz|. $$ Here $\omega_h(0,\cdot;D)$ is the discrete harmonic measure at $0$ for the random walk, and $\omega(0,\cdot;D)$ is the (continuous) harmonic measure at $0$. We give an explicit formula for $\sigma_D$ in terms of the conformal map from $D$ to the unit disc. The proof relies on some fine approximations of the potential kernel and Green's function of the random walk by their continuous counterparts, which may be of independent interest.