The difference between a discrete and continuous harmonic measure

2015/06/13 by Jianping Jiang, Jiang, Jianping, Tom Kennedy +1
Computer Science · Mathematics · #60G50 #60J65 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals #Probability (math.PR) #math.PR #msc:60G50 #msc:60J65

paper · pdf · doi:10.48550/arxiv.1506.04313

16 pages, revision after the referee's report, to appear in Journal of Theoretical Probability

openalex publication_date 2015/06/13 · arxiv created 2016/05/27 · arxiv updated 2016/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ω_h(0,⋅;D) be the discrete harmonic measure at 0∈ D associated with this random walk, and ω(0,⋅;D) be the (continuous) harmonic measure at 0. For domains D with analytic boundary, we prove there is a bounded continuous function σ_D(z) on ∂ D such that for functions g which are in C^2+α(∂ D) for some α>0 lim_{h\downarrow 0} \frac∫_{∂ D} g(ξ) ω_h(0,|dξ|;D) -∫_{∂ D} g(ξ)ω(0,|dξ|;D)h = ∫_{∂ D}g(z) σ_D(z) |dz|. We give an explicit formula for σ_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.

Citations

• �ber die partiellen Differenzengleichungen der mathematischen Physik

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