Stochastic billiards on general tables
type: Item
instance of: scholarly article
title: Stochastic billiards on general tables
zbMATH Open document ID: 1015.60058
author: Steven Neil Evans
published in: The Annals of Applied Probability
publication date: 2003-05-06T00:00:00Z
review text: By definition, billiards are dynamical systems that model the motion of a particle in some \(d\)-dimensional region (the ``table'') with conditions on the nature of ``reflections'' at the boundary. In the most classical case, the particle moves with unit speed and constant direction in the interior of the ``table'' and new directions are chosen at the boundary via the usual ``angle of incident equals angle of reflection'' rule. The author considers stochastic analogs with dynamics that can be described as follows: a particle moves with unit speed and constant direction in the interior of the ``table'' until it strikes the boundary. When the particle strikes the boundary, a new direction for the particle is chosen uniformly at random from the hemisphere of directions that point into the interior of the region and the motion continues. The equilibrium distribution of such stochastic billiard processes is investigated and the manner in which these processes converge to equilibrium is studied. The discrete Markov chain records the locations of successive hits on the boundary. It is shown that, uniformly in the starting point, there is exponentially fast total variation convergence to an invariant distribution. By analyzing an associated nonlinear, first-order PDE, it is investigated which regions are such that chain is reversible with respect to surface measure on the boundary. Corresponding results are considered for polygonal regions in the plane.
Mathematics Subject Classification ID: 60J05
Mathematics Subject Classification ID: 37D50
Mathematics Subject Classification ID: 37H99
Mathematics Subject Classification ID: 60J25
zbMATH DE Number: 1906248
zbMATH Keywords: billiard
zbMATH Keywords: Markov chain
zbMATH Keywords: Markov process
zbMATH Keywords: ergodic
zbMATH Keywords: irreducible
zbMATH Keywords: coupling
zbMATH Keywords: shift-coupling
zbMATH Keywords: total variation
zbMATH Keywords: reversibility
zbMATH Keywords: shake and bake
reviewed by: Anatoliy Swishchuk
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1015345298
OpenAlex ID: W1969595341
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DOI: 10.1214/AOAP/1015345298
description: scientific article; zbMATH DE number 1906248