Preface ........................................................ ix
Chapter 1 Basics on large deviations ........................... 1
1.1 Gartner-Ellis theorem ...................................... 1
1.2 LDP for non-negative random variables ...................... 8
1.3 LDP by sub-additivity ..................................... 19
1.4 Notes and comments ........................................ 22
Chapter 2 Brownian intersection local times ................... 25
2.1 Introduction .............................................. 25
2.2 Mutual intersection local time ............................ 27
2.3 Self-intersection local time .............................. 42
2.4 Renormalization ........................................... 48
2.5 Notes and comments ........................................ 53
Chapter 3 Mutual intersection: large deviations ............... 59
3.1 High moment asymptotics ................................... 59
3.2 High moment of α([0, τ1] × • • • × [0, τp]) ............... 67
3.3 Large deviation for α([0,1]p) ............................. 77
3.4 Notes and comments ........................................ 84
Chapter 4 Self-intersection: large deviations ................. 91
4.1 Feynman-Kac formula ....................................... 91
4.2 One-dimensional case ..................................... 102
4.3 Two-dimensional case ..................................... 111
4.4 Applications to LIL ...................................... 121
4.5 Notes and comments ....................................... 126
Chapter 5 Intersections on lattices: weak convergence ........ 133
5.1 Preliminary on random walks .............................. 133
5.2 Intersection in 1-dimension .............................. 139
5.3 Mutual intersection in sub-critical dimensions ........... 145
5.4 Self-intersection in dimension two ....................... 160
5.5 Intersection in high dimensions .......................... 164
5.6 Notes and comments ....................................... 171
Chapter 6 Inequalities and integrabilities ................... 177
6.1 Multinomial inequalities ................................. 177
6.2 Integrability of In and Jn ............................... 187
6.3 Integrability of Qn and Rn in low dimensions ............. 191
6.4 Integrability of Qn and Rn in high dimensions ............ 198
6.5 Notes and comments ....................................... 204
Chapter 7 Independent random walks: large deviations ......... 207
7.1 Feynman-Kac minorations .................................. 207
7.2 Moderate deviations in sub-critical dimensions ........... 222
7.3 Laws of the iterated logarithm ........................... 226
7.4 What do we expect in critical dimensions? ................ 230
7.5 Large deviations in super-critical dimensions ............ 231
7.6 Notes and comments ....................................... 247
Chapter 8. Single random walk: large deviations ............. 253
8.1 Self-intersection in one dimension ....................... 253
8.2 Self-intersection in d = 2 ............................... 257
8.3 LDP of Gaussian tail in d = 3 ............................ 264
8.4 LDP of non-Gaussian tail in d = 3 ........................ 270
8.5 LDP for renormalized range in d = 2,3 .................... 278
8.6 Laws of the iterated logarithm ........................... 287
8.7 What do we expect in d ≥ 4? .............................. 289
8.8 Notes and comments ....................................... 291
Appendix ...................................................... 297
A Green's function ........................................... 297
B Fourier transformation ..................................... 299
C Constant k(d,p) and related variations ..................... 303
D Regularity of stochastic processes ......................... 309
E Self-adjoint operators ..................................... 313
Bibliography .................................................. 321
List of General Notations ..................................... 329
Index ......................................................... 331
|
