The Annals of Applied Probability

Stochastic billiards on general tables

Steven N. Evans

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We consider stochastic analogs of classical billiard systems. A particle moves at unit speed with constant direction in the interior of a bounded, d-dimensional region with continuously differentiable boundary. The boundary need not be connected; that is, the “table” may have inte- rior “obstacles.” When the particle strikes the boundary, a new direction is chosen uniformly at random from the directions that point back into the interior of the region and the motion continues. Such chains are closely related to those that appear in shake-and-bake simulation algorithms. For the discrete time Markov chain that records the locations of successive hits on the boundary, we show 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, we investigate which regions are such that this chain is reversible with respect to surface measure on the boundary. We also establish a result on uniform total variation Césaro convergence to equilibrium for the continuous time Markov process that tracks the position and direction of the particle. A key ingredient in our proof is a result on the geometry of $C^1$ regions that can be described loosely as follows:associated with any bounded $C^1$ region is an integer N such that it is always possible to pass a message between any two locations in the region using a relay of exactly N locations with the property that every location in the relay is directly visible from its predecessor. Moreover, the locations of the intermediaries can be chosen from a fixed, finite subset of positions on the boundary of the region. We also consider corresponding results for polygonal regions in the plane.

Article information

Ann. Appl. Probab., Volume 11, Number 2 (2001), 419-437.

First available in Project Euclid: 5 March 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J05: Discrete-time Markov processes on general state spaces 60J25: Continuous-time Markov processes on general state spaces
Secondary: 34C35 58F11

Billiards Markov chain Markov process ergodic irreducible coupling shift-coupling total variation reversibility shake-and-bake


Evans, Steven N. Stochastic billiards on general tables. Ann. Appl. Probab. 11 (2001), no. 2, 419--437. doi:10.1214/aoap/1015345298.

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