ISBN: 3-540-66593-5
TITLE: Lectures on Probability Theory and Statistics
AUTHOR: Bertoin, J.; Martinelli, F.; Peres, Y.
TOC:

Jean BERTOIN : "SUBORDINATORS: EXAMPLES AND APPLICATIONS"
0. Foreword 4
1. Elements on subordinators 5
1.1 Definitions and first properties 5
1.2 The Lvy-Khintchine formula 7
1.3 The renewal measure 9
1.4 The range of a subordinator 13
2. Regenerative property 17
2.1 Regenerative sets 17
2.2 Connection with Markov processes 19
3. Asymptotic behaviour of last passage times 23
3.1 Asymptotic behaviour in distribution 23
3.1.1 The self-similar case 23
3.1.2 The Dynkin-Lamperti theorem 24
3.2 Asymptotic sample path behaviour 27
4. Rates of growth of local time 31
4.1 Law of the iterated logarithm 31
4.2 Modulus of continuity 35
5. Geometric properties of regenerative sets 39
5.1 Fractal dimensions 39
5.1.1 Box-counting dimension 39
5.1.2 Hausdorff and packing dimensions 41
5.2 Intersections with a regenerative set 43
5.2.1 Equilibrium measure and capacity 43
5.2.2 Dimension criteria 45
5.2.3 Intersection of independant regenerative sets 47
6. Burgers equation with Brownian initial velocity 50
6.1 Burgers equation and the Hopf-Cole solution 50
6.2 Brownian initial velocity 51
6.3 Proof of the theorem 52
7. Random covering 55
7.1 Setting 55
7.2 The Laplace exponent of the uncovered set 57
7.3 Some properties of the uncovered set 57
8. Lvy processes 62
8.1 Local time at a fixed point 62
8.2 Local time at the supremum 65
8.3 The spectrally negative case 67
8.4 Bochner's subordination for Lvy processes 69
9. Occupation times of a linear Brownian motion 73
9.1 Occupation times and subordinators 73
9.2 Lvy measure and Laplace exponent 74
9.2.1 Lvy measure via excursion theory 74
9.2.2 Laplace exponent via the Sturm-Liouville equation 75
9.2.3 Spectral representation of the Laplace exponent 77
9.3 The zero set of a one-dimensional diffusion 79
Fabio MARTINELLI: "LECTURES ON GLAUBER DYNAMICS FOR DISCRETE SPIN MODELS"
1. Introduction 96
2. Gibbs Measures of Lattice Spin Models 98
2.1 Notation 98
2.2 Gibbs Measures 99
2.3 Weak and String Mixing Conditions 100
2.4 Mixing properties and bounds on relative densities 102
3. The Glauber Dynamics 110
3.1 The Dynamics in Finite Volume 110
3.2 Infinite Volume Dynamics 110
3.3 Graphical Construction 111
3.4 Attractive Dynamics for Ferromagnetic Interactions 113
3.5 Spectral Gap and Logarithmic Sobolev Constant 114
3.6 From Single Spin Dynamics to Block Dynamics 118
3.7 General Results on the Spectral Gap 119
3.8 General Results on the Logarithmic Sobolev Constant 124
3.9 Possible Rates of Convergence to Equilibrium 125
4. One Phase Region 130
4.1 The Attractive Case 130
4.2 The General Case : Recursive Analysis 134
5. Boundary Phase Transitions 144
5.1 The Solid-on-Solid Approximation 144
5.2 Back to the Ising-Model 148
5.3 Recent Progresses 149
6. Phase Coexistence 151
6.1 Some Preliminary Key Equilibrium Results 152
6.2 A Geometric Bound on the Spectral Gap 155
6.3 A Lower Bound on the Spectral Gap with + B.C 158
6.4 A Lower Bound on the Spectral Gap with Free B.C 161
6.5 Upper Bound on the Spectral Gap with Free B.C 163
6.6 Mixed B.C 164
6.7 Applications 164
7. Glauber Dynamics for the Dilute Ising Model 170
7.1 The Dynamics in the Paramagnetic Phase 171
7.2 The Dynamics in the Griffiths Phase: p<p_c 171
7.3 The Dynamics in the Griffiths Phase: p>p_c 175
7.4 A Coarse Grained Description Above p_c 178
7.5 Proof of the Main Results Above p_c 181
Yuval PERES : "PROBABILITY ON TREES: AN INTRODUCTORY CLIMB"
1. Preface 195
2. Basic Definitions and a Few Highlights 197
3. Galton-Watson Trees 203
4. General percolation on a connected graph 204
5. The First-Moment Method 206
6. Quasi-independent Percolation 209
7. The Second Moment Method 210
8. Electrical Networks 213
9. Infinite Networks 219
10. The Method of Random Paths 221
11. Transience of Percolation Clusters 223
12. Subperiodic Trees 228
13. The Random Walks RW 230
14. Capacity 231
15. Intersection-Equivalence 238
16. Reconstruction for the Ising Model on a Tree 243
17. Unpredictable Paths in Z and EIT in Z3 255
18. Tree-Indexed Processes 260
19. Recurrence for Tree-Indexed Markov Chains 265
20. Dynamical Percolation 266
21. Stochastic Domination Between Trees 272
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