The difference between a discrete and continuous harmonic measure

TL;DR

This paper investigates the difference between discrete and continuous harmonic measures for a random walk in a planar domain, providing an explicit formula for their difference's asymptotic behavior as the step size tends to zero.

Contribution

It establishes a precise asymptotic expansion for the difference between discrete and continuous harmonic measures, including an explicit formula for the correction term.

Findings

Derived an explicit formula for the correction function  in terms of conformal maps.

Proved the asymptotic limit of the difference between measures as step size approaches zero.

Used potential kernel and Green's function approximations for the analysis.

Abstract

We consider a discrete-time, continuous-state random walk with steps uniformly distributed in a disk of radius of $h$. For a simply connected domain $D$ in the plane, let $\omega_h(0,\cdot;D)$ be the discrete harmonic measure at $0\in D$ associated with this random walk, and $\omega(0,\cdot;D)$ be the (continuous) harmonic measure at $0$. For domains $D$ with analytic boundary, we prove there is a bounded continuous function $\sigma_D(z)$ on $\partial D$ such that for functions $g$ which are in $C^{2+\alpha}(\partial D)$ for some $\alpha>0$ $$ \lim_{h\downarrow 0} \frac{\int_{\partial D} g(\xi) \omega_h(0,|d\xi|;D) -\int_{\partial D} g(\xi)\omega(0,|d\xi|;D)}{h} = \int_{\partial D}g(z) \sigma_D(z) |dz|. $$ 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.