Random walks are stochastic processes formed by successive summation of independent, identically distributed random variables and are one of the most studied topics in probability theory. This contemporary introduction evolved from courses taught at Cornell University and the University of Chicago by the first author, who is one of the most highly regarded researchers in the field of stochastic processes. This text meets the need for a modern reference to the detailed properties of an important class of random walks on the integer lattice. It is suitable for probabilists, mathematicians working in related fields, and for researchers in other disciplines who use random walks in modeling.
Author(s): Gregory F. Lawler, Vlada Limic
Series: Cambridge Studies in Advanced Mathematics 123
Publisher: CUP
Year: 2010
Language: English
Pages: 378
Tags: Математика;Теория вероятностей и математическая статистика;Теория случайных процессов;
Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
1.1 Basic definitions......Page 15
1.2 Continuous-time random walk......Page 20
1.3 Other lattices......Page 21
1.5 Generator......Page 25
1.6 Filtrations and strong Markov property......Page 28
1.7 A word about constants......Page 31
Exercises......Page 32
2.1 Introduction......Page 35
2.2.1 Characteristic functions of random variables in Rd......Page 39
2.2.2 Characteristic functions of random variables in Zd......Page 41
2.3 LCLT – characteristic function approach......Page 42
2.3.1 Exponential moments......Page 60
2.4 Some corollaries of the LCLT......Page 65
2.5.1 Stirling’s formula and one-dimensional walks......Page 72
2.5.2 LCLT for Poisson and continuous-time walks......Page 78
Exercises......Page 82
3.1 Introduction......Page 86
3.2 Construction of Brownian motion......Page 88
3.3 Skorokhod embedding......Page 93
3.4 Higher dimensions......Page 96
3.5 An alternative formulation......Page 98
Exercises......Page 99
4.1 Recurrence and transience......Page 101
4.2 The Green’s generating function......Page 102
4.3 The Green’s function, transient case......Page 109
4.3.1 Asymptotics under weaker assumptions......Page 113
4.4.1 Two dimensions......Page 115
4.4.2 Asymptotics under weaker assumptions......Page 121
4.4.3 One dimension......Page 123
4.5 Fundamental solutions......Page 127
4.6 The Green’s function for a set......Page 128
Exercises......Page 134
5.1 Gambler’s ruin estimate......Page 137
5.1.1 General case......Page 141
5.2 One-dimensional killed walks......Page 149
5.3 Hitting a half-line......Page 152
Exercises......Page 157
6.1 Introduction......Page 158
6.2 Dirichlet problem......Page 160
6.3 Difference estimates and Harnack inequality......Page 166
6.4 Further estimates......Page 174
6.5 Capacity, transient case......Page 180
6.6 Capacity in two dimensions......Page 190
6.7 Neumann problem......Page 200
6.8 Beurling estimate......Page 203
6.9 Eigenvalue of a set......Page 208
Exercises......Page 213
7.1 Introduction......Page 219
7.2 Some estimates......Page 221
7.3 Quantile coupling......Page 224
7.4 The dyadic coupling......Page 227
7.5 Proof of Theorem 7.1.1......Page 230
7.6 Higher dimensions......Page 232
7.7 Coupling the exit distributions......Page 233
Exercises......Page 236
8.1 Poisson kernel......Page 239
8.1.1 Half space......Page 240
8.1.2 Cube......Page 243
8.1.3 Strips and quadrants in Z2......Page 249
8.2 Eigenvalues for rectangles......Page 252
8.3 Approximating continuous harmonic functions......Page 253
8.4 Estimates for the ball......Page 255
Exercises......Page 258
9.2 Definitions and notations......Page 261
9.2.1 Simple random walk on a graph......Page 265
9.3 Generating functions and loop measures......Page 266
9.4 Loop soup......Page 271
9.5 Loop erasure......Page 273
9.6 Boundary excursions......Page 275
9.7 Wilson’s algorithm and spanning trees......Page 282
9.8.1 Complete graph......Page 285
9.8.2 Hypercube......Page 286
9.8.3 Sierpinski graphs......Page 289
9.9 Spanning trees of subsets of Z2......Page 291
9.10 Gaussian free field......Page 303
Exercises......Page 308
10.1 Long-range estimate......Page 311
10.2 Short-range estimate......Page 316
10.3 One-sided exponent......Page 319
11.1 h-processes......Page 321
11.2 Loop-erased random walk......Page 325
11.3 LERWin Zd......Page 327
11.3.1 d ≥ 3......Page 328
11.3.2 d = 2......Page 329
11.4 Rate of growth......Page 333
11.5 Short-range intersections......Page 337
Exercises......Page 338
A.1.1 Riemann sums......Page 340
A.1.2 Logarithm......Page 341
A.2 Martingales......Page 345
A.2.1 Optional sampling theorem......Page 346
A.2.2 Maximal inequality......Page 348
A.2.3 Continuous martingales......Page 350
A.3 Joint normal distributions......Page 351
A.4 Markov chains......Page 353
A.4.1 Chains restricted to subsets......Page 356
A.4.2 Maximal coupling of Markov chains......Page 360
A.5 Some Tauberian theory......Page 365
A.6 Second moment method......Page 367
A.7 Subadditivity......Page 368
Exercises......Page 369
Bibliography......Page 374
Index of Symbols......Page 375
Index......Page 377
