Abstract
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
Citation
Steven N. Evans. "Stochastic billiards on general tables." Ann. Appl. Probab. 11 (2) 419 - 437, May 2001. https://doi.org/10.1214/aoap/1015345298
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