Some Estimates for Planar Random Walk and Brownian Motion

2006/11/06 by Christian Beneš, Christian Benes, Benes, Christian
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #60G50 #60J65 #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #msc:60G50 #msc:60J65

paper · pdf · doi:10.48550/arxiv.math/0611127

18 pages, 2 figures

arxiv created 2006/11/06 · openalex publication_date 2006/11/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The purpose of this note is to collect in one place a few results about simple random walk and Brownian motion which are often useful. These include standard results such as Beurling estimates, large deviation estimates, and a method for coupling the two processes, as well as solutions to the discrete Dirichlet problem in various domains which, to the author's knowledge, have not been published anywhere. The main focus is on the two-dimensional processes.

Related

• The Beurling estimate for a class of random walks
• Convergence in natural parametrization of random walk frontier
• The Characteristic Function of the Renormalized Intersection Local Time of the Planar Brownian Motion
• The joint distribution of value and local time of simple random walk and reflected simple random walk
• Critical exponents, conformal invariance and planar Brownian motion