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 h. For a simply connected domain D in the plane, let omega(h)(0, .; D) be the discrete harmonic measure at 0 is an element of D associated with this random walk, and omega(0, .; 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 derivative D such that for functions g which are in C2+sigma (partial derivative D) for some alpha > 0 we have lim(h down arrow 0) integral(partial derivative D) g(xi)omega(h)( 0, vertical bar d xi vertical bar; D) - integral(partial derivative D) g(xi)omega(0, vertical bar d xi vertical bar; D)/h = integral(partial derivative D) g(z)sigma(D)(z)vertical bar dz vertical bar. We give an explicit formula for sigma(D) in terms of the conformal map from D to the unit disk. 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.