The Difference Between a Discrete and Continuous Harmonic Measure

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 $$\omega _h(0,\cdot ;D)$$ ω h ( 0 , · ; D ) be the discrete harmonic measure at $$0\in D$$ 0 ∈ D associated with this random walk, and $$\omega (0,\cdot ;D)$$ ω ( 0 , · ; D ) be the (continuous) harmonic measure at 0. For domains D with analytic boundary, we prove there is a bounded continuous function $$\sigma _D(z)$$ σ D ( z ) on $$\partial D$$ ∂ D such that for functions g which are in $$C^{2+\alpha }(\partial D)$$ C 2 + α ( ∂ D ) for some $$\alpha >0$$ α > 0 we have $$\begin{aligned} \lim _{h\downarrow 0} \frac{\int _{\partial D} g(\xi ) \omega _h(0,|\mathrm{d}\xi |;D) -\int _{\partial D} g(\xi )\omega (0,|\mathrm{d}\xi |;D)}{h} = \int _{\partial D}g(z) \sigma _D(z) |\mathrm{d}z|. \end{aligned}$$ lim h ↓ 0 ∫ ∂ D g ( ξ ) ω h ( 0 , | d ξ | ; D ) - ∫ ∂ D g ( ξ ) ω ( 0 , | d ξ | ; D ) h = ∫ ∂ D g ( z ) σ D ( z ) | d z | . We give an explicit formula for $$\sigma _D$$ σ 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.