Contents
Preface ix
Chapter 1. Basics on large deviations 1
1.1. artner-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
vii
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