## 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 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 we have (Formula Presented.)We give an explicit formula for 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.

Original language | English (US) |
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Pages (from-to) | 1424-1444 |

Number of pages | 21 |

Journal | Journal of Theoretical Probability |

Volume | 30 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 2017 |

## Keywords

- Brownian motion
- Dirichlet problem
- Harmonic measure
- Random walk

## ASJC Scopus subject areas

- Statistics and Probability
- General Mathematics
- Statistics, Probability and Uncertainty