| Subordinators Examples and Applications | |
| Foreword | p. 4 |
| Elements on subordinators | p. 5 |
| Definitions and first properties | p. 5 |
| The Lévy-Khintchine formula | p. 7 |
| The renewal measure | p. 9 |
| The range of a subordinator | p. 13 |
| Regenerative property | p. 17 |
| Regenerative sets | p. 17 |
| Connection with Markov processes | p. 19 |
| Asymptotic behaviour of last passage times | p. 23 |
| Asymptotic behaviour in distribution | p. 23 |
| The self-similar case | p. 23 |
| The Dynkin-Lamperti theorem | p. 24 |
| Asymptotic sample path behaviour | p. 27 |
| Rates of growth of local time | p. 31 |
| Law of the iterated logarithm | p. 31 |
| Modulus of continuity | p. 35 |
| Geometric properties of regenerative sets | p. 39 |
| Fractal dimensions | p. 39 |
| Box-counting dimension | p. 39 |
| Hausdorff and packing dimensions | p. 41 |
| Intersections with a regenerative set | p. 43 |
| Equilibrium measure and capacity | p. 43 |
| Dimension criteria | p. 45 |
| Intersection of independant regenerative sets | p. 47 |
| Burgers equation with Brownian initial velocity | p. 50 |
| Burgers equation and the Hopf-Cole solution | p. 50 |
| Brownian initial velocity | p. 51 |
| Proof of the theorem | p. 52 |
| Random covering | p. 55 |
| Setting | p. 55 |
| The Laplace exponent of the uncovered set | p. 57 |
| Some properties of the uncovered set | p. 57 |
| Levy processes | p. 62 |
| Local time at a fixed point | p. 62 |
| Local time at the supremum | p. 65 |
| The spectrally negative case | p. 67 |
| Bochner's subordination for Lévy processes | p. 69 |
| Occupation times of a linear Brownian motion | p. 73 |
| Occupation times and subordinators | p. 73 |
| Levy measure and Laplace exponent | p. 74 |
| Lévy measure via excursion theory | p. 74 |
| Laplace exponent via the Sturm-Liouville equation | p. 75 |
| Spectral representation of the Laplace exponent | p. 77 |
| The zero set of a one-dimensional diffusion | p. 79 |
| Lectures on Glauber Dynamics for Discrete Spin Models | |
| Introduction | p. 96 |
| Gibbs Measures of Lattice Spin Models | p. 98 |
| Notation | p. 98 |
| Gibbs Measures | p. 99 |
| Weak and String Mixing Conditions | p. 100 |
| Mixing properties and bounds on relative densities | p. 102 |
| The Glauber Dynamics | p. 110 |
| The Dynamics in Finite Volume | p. 110 |
| Infinite Volume Dynamics | p. 110 |
| Graphical Construction | p. 111 |
| Attractive Dynamics for Ferromagnetic Interactions | p. 113 |
| Spectral Gap and Logarithmic Sobolev Constant | p. 114 |
| From Single Spin Dynamics to Block Dynamics | p. 118 |
| General Results on the Spectral Gap | p. 119 |
| General Results on the Logarithmic Sobolev Constant | p. 124 |
| Possible Rates of Convergence to Equilibrium | p. 125 |
| One Phase Region | p. 130 |
| The Attractive Case | p. 130 |
| The General Case Recursive Analysis | p. 134 |
| Boundary Phase Transitions | p. 144 |
| The Solid-on-Solid Approximation | p. 144 |
| Back to the Ising-Model | p. 148 |
| Recent Progresses | p. 149 |
| Phase Coexistence | p. 151 |
| Some Preliminary Key Equilibrium Results | p. 152 |
| A Geometric Bound on the Spectral Gap | p. 155 |
| A Lower Bound on the Spectral Gap with + B.C | p. 158 |
| A Lower Bound on the Spectral Gap with Free B.C | p. 161 |
| Upper Bound on the Spectral Gap with Free B.C | p. 163 |
| Mixed B.C | p. 164 |
| Applications | p. 164 |
| Glauber Dynamics for the Dilute Ising Model | p. 170 |
| The Dynamics in the Paramagnetic Phase | p. 171 |
| The Dynamics in the Griffiths Phase p < pc | p. 171 |
| The Dynamics in the Griffiths Phase p = pc | p. 175 |
| A Coarse Grained Description Above pc | p. 178 |
| Proof of the Main Results Above pc | p. 181 |
| Probability on Trees an Introductory Climb | |
| Preface | p. 195 |
| Basic Definitions and a Few Highlights | p. 197 |
| Galton-Watson Trees | p. 203 |
| General percolation on a connected graph | p. 204 |
| The First-Moment Method | p. 206 |
| Quasi-independent Percolation | p. 209 |
| The Second Moment Method | p. 210 |
| Electrical Networks | p. 213 |
| Infinite Networks | p. 219 |
| The Method of Random Paths | p. 221 |
| Transience of Percolation Clusters | p. 223 |
| Subperiodic Trees | p. 228 |
| The Random Walks RW | p. 230 |
| Capacity | p. 231 |
| Intersection-Equivalence | p. 238 |
| Reconstruction for the Ising Model on a Tree | p. 243 |
| Unpredictable Paths in Z and EIT in Z3 | p. 255 |
| Tree-Indexed Processes | p. 260 |
| Recurrence for Tree-Indexed Markov Chains | p. 265 |
| Dynamical Percolation | p. 266 |
| Stochastic Domination Between Trees | p. 272 |
| Table of Contents provided by Publisher. All Rights Reserved. |
