Memoirs of the American Mathematical Society 2009; 193 pp; softcover Volume: 198 ISBN10: 082184282X ISBN13: 9780821842829 List Price: US$78 Individual Members: US$46.80 Institutional Members: US$62.40 Order Code: MEMO/198/927
 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 the authors 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. The authors 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. The 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. Table of Contents  Introduction
 Statement of results
 Plan of the proofs
 Standard pairs and equidistribution
 Regularity of the diffusion matrix
 Moment estimates
 Fast slow particle
 Small large particle
 Open problems
 Appendix A. Statistical properties of dispersing billiards
 Appendix B. Growth and distortion properties of dispersing billiards
 Appendix C. Distortion bounds for two particle system
 Bibliography
 Index
