Stochastic billiards on general tables
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.
scientific article; zbMATH DE number 1906248
scientific article