  In this paper, we are interested in the speed of convergence of the
stochastic billiard evolving in a convex set K. This process can be described
as follows: a particle moves at unit speed inside the set K until it hits the
boundary, and is randomly reflected, independently of its position and previous
velocity. We focus on convex sets in R 2 with a curvature bounded from above
and below. We give an explicit coupling for both the continuous-time process
and the embedded Markov chain of hitting points on the boundary, which leads to
an explicit speed of convergence to equilibrium.

Probability