One Dimensional Random Walk with a Partially Reflecting Barrier

1963/06/01 by G. Lehner · 2 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Combinatorics #Diffusion and Search Dynamics #Hypergeometric distribution #Hypergeometric function #Lattice (music) #Mathematical analysis #Mathematics #Optics #Physics #Pure mathematics #Random walk #Reflection (computer programming) #Reflection coefficient #Scientific Research and Discoveries #Statistical physics #Statistics #Stochastic processes and statistical mechanics

paper · pdf · doi:10.1214/aoms/1177704151

openalex publication_date 1963/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2025/11/06

Abstract

In the present paper we consider the one dimensional random walk of a particle restricted by a partially reflecting barrier. The reflecting barrier is described by a coefficient of reflection r. The probability of finding a particle at a lattice point m after N steps is calculated and expressed in terms of hypergeometric functions of the ₂F₁-type. Other theorems are deduced concerning the one dimensional random walk. For instance the number of paths leading from one lattice point to another lattice point in N steps and showing a given number of reflections at the barrier is calculated.

Cited by

• Bi-Directional Grid Constrained Stochastic Processes' Link to Multi-Skew Brownian Motion
• Diffusion Through a Network of Compartments Separated by Partially-Transmitting Boundaries

Related

• One dimensional lattice random walks with absorption at a point / on a half line
• Method of images for one-dimensional discrete random walk under a reflecting barrier
• Random walk on a diagonal lattice with absorbing boundaries
• Boundary contacts for reflected random walks in the quarter plane
• Random walks and Brownian motion: A method of computation for first-passage times and related quantities in confined geometries