The difference between a discrete and continuous harmonic measure

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.