# Brownian Brownian Motion-I

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*N. Chernov; D. Dolgopyat*

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

# Table of Contents

## Brownian Brownian Motion-I

- Contents v6 free
- Chapter 1. Introduction 19 free
- Chapter 2. Statement of results 715
- Chapter 3. Plan of the proofs 1523
- Chapter 4. Standard pairs and equidistribution 2533
- Chapter 5. Regularity of the diffusion matrix 5765
- 5.1. Transport coefficients 5765
- 5.2. Reduction to a finite series 6270
- 5.3. Integral estimates: general scheme 6472
- 5.4. Integration by parts 6977
- 5.5. Cancellation of large boundary terms 7785
- 5.6. Estimation of small boundary terms 8189
- 5.7. Two-sided integral sums 8492
- 5.8. Bounding off-diagonal terms 8896
- 5.9. Hölder approximation 9098

- Chapter 6. Moment estimates 93101
- 6.1. General plan 93101
- 6.2. Structure of the proofs 98106
- 6.3. Short term moment estimates for V 100108
- 6.4. Moment estimates–a priori bounds 103111
- 6.5. Tightness 110118
- 6.6. Second moment 113121
- 6.7. Martingale property 115123
- 6.8. Transition to continuous time 117125
- 6.9. Uniqueness for stochastic differential equations 118126

- Chapter 7. Fast slow particle 123131
- Chapter 8. Small large particle 129137
- Chapter 9. Open problems 133141
- Appendix A. Statistical properties of dispersing billiards 139147
- Appendix B. Growth and distortion in dispersing billiards 167175
- Appendix C. Distortion bounds for two particle system 177185
- Bibliography 187195
- Index 193201 free