Source: Ann. Appl. Probab.
Volume 11, Number 2
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.
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