 | Brownian Brownian motion-I / Chernov N., Dolgopyat D. - Providence: AMS, 2009. - vii, 193 p.: ill. - (Memoirs of the American Mathematical Society; Vol.198, N 927) - ISBN 978-0-8218-4282-9; ISSN 0065-9266
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Chapter 1. Introduction ......................................... 1
1.1. The model .................................................. 1
1.2. The container .............................................. 2
1.3. Billiard approximations .................................... 3
Chapter 2. Statement of results ................................. 7
2.1. Heavy disk in 'equilibrium' (linear motion) ................ 7
2.2. Heavy disk at rest (slow acceleration) ..................... 9
2.3. Heavy disk of small size .................................. 11
2.4. Comparison to previous works .............................. 12
Chapter 3. Plan of the proofs .................................. 15
3.1. General strategy .......................................... 15
3.2. Precise definitions ....................................... 17
3.3. Key technical results ..................................... 20
Chapter 4. Standard pairs and equidistribution ................. 25
4.1. Unstable vectors .......................................... 25
4.2. Unstable curves ........................................... 29
4.3. Homogeneous unstable curves ............................... 33
4.4. Standard pairs ............................................ 36
4.5. Perturbative analysis ..................................... 42
4.6. Equidistribution properties ............................... 47
Chapter 5. Regularity of the diffusion matrix .................. 57
5.1. Transport coefficients .................................... 57
5.2. Reduction to a finite series .............................. 62
5.3. Integral estimates: general scheme ........................ 64
5.4. Integration by parts ...................................... 69
5.5. Cancellation of large boundary terms ...................... 77
5.6. Estimation of small boundary terms ........................ 81
5.7. Two-sided integral sums ................................... 84
5.8. Bounding off-diagonal terms ............................... 88
5.9. Holder approximation ...................................... 90
Chapter 6. Moment estimates .................................... 93
6.1. General plan .............................................. 93
6.2. Structure of the proofs ................................... 98
6.3. Short term moment estimates for V ........................ 100
6.4. Moment estimates-a priori bounds ......................... 103
6.5. Tightness ................................................ 110
6.6. Second moment ............................................ 113
6.7. Martingale property ...................................... 115
6.8. Transition to continuous time ............................ 117
6.9. Uniqueness for stochastic differential equations ......... 118
Chapter 7. Fast slow particle ................................. 123
Chapter 8. Small large particle ............................... 129
Chapter 9. Open problems ...................................... 133
9.1. Collisions of the massive disk with the wall ............. 133
9.2. Longer time scales ....................................... 133
9.3. Stadia and the piston problem ............................ 133
9.4. Finitely many particles .................................. 134
9.5. Growing number of particles .............................. 135
9.6. Particles of positive size ............................... 136
Appendix A. Statistical properties of dispersing
billiards ......................................... 139
A.l. Decay of correlations: overview ................... 139
A.2. Decay of correlations: extensions ................. 149
A.3. Large deviations .................................. 153
A.4. Moderate deviations ............................... 154
A.5. Nonsingularity of diffusion matrix ................ 158
A.6. Asymptotics of diffusion matrix ................... 159
Appendix B. Growth and distortion in dispersing
billiards ......................................... 167
B.l. Regularity of H-curves ............................ 167
B.2. Invariant Section Theorem ......................... 171
B.3. The function space  ............................. 174
Appendix C. Distortion bounds for two particle system ......... 177
Bibliography .................................................. 187
Index ......................................................... 193
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