# Explicit speed of convergence of the stochastic billiard in a convex set

## TL;DR

This paper analyzes the convergence speed of a stochastic billiard process in convex sets, providing explicit coupling methods and convergence rates for both continuous and discrete boundary hitting processes.

## Contribution

It introduces explicit coupling techniques to determine the convergence rate of stochastic billiards in convex sets with bounded curvature.

## Key findings

- Explicit convergence rates derived for the stochastic billiard process.
- Coupling methods applicable to both continuous and discrete boundary hitting times.
- Results depend on curvature bounds of the convex set.

## Abstract

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.