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Brownian Brownian Motion-I
N. Chernov, University of Alabama, Birmingham, AL, and D. Dolgopyat, University of Maryland, College Park, MD
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Memoirs of the American Mathematical Society
2009; 193 pp; softcover
Volume: 198
ISBN-10: 0-8218-4282-X
ISBN-13: 978-0-8218-4282-9
List Price: US$78
Individual Members: US$46.80
Institutional Members: US$62.40
Order Code: MEMO/198/927
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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
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