### Brownian Brownian Motion

Nikolai Chernov

Abstract. A classical model of Brownian motion consists of a
heavy molecule submerged into a gas of light atoms in a closed
container. In this work we study a 2D version of this model,
where the molecule is a heavy disk of mass $M \gg 1$ and the
gas is represented by just one point particle of mass $m=1$,
which interacts with the disk and the walls of the container
via elastic collisions. Chaotic behavior of the particles is
ensured by convex (scattering) walls of the container. We prove
that the position and velocity of the disk, in an appropriate
time scale, converge, as $M\to\infty$, to a Brownian motion
(possibly, inhomogeneous); the scaling regime and the structure
of the limit process depend on the initial conditions. Our
proofs are based on strong hyperbolicity of the underlying
dynamics, fast decay of correlations in systems with elastic
collisions (billiards), and methods of averaging theory.