Markov processes, semigroups and generators

✍ Scribed by Kolokoltsov V.N.

• Publisher: De Gruyter
• Year: 2011
• Tongue: English
• Leaves: 449
• Series: De Gruyter Studies in Mathematics
• Category: Library

✦ Synopsis

This work offers a highly useful, well developed reference on Markov processes, the universal model for random processes and evolutions. The wide range of applications, in exact sciences as well as in other areas like social studies, require a volume that offers a refresher on fundamentals before conveying the Markov processes and examples for applications. This work does just that, and with the necessary mathematical rigor.

✦ Table of Contents

Preface......Page 8
Notations......Page 12
Standard abbreviations......Page 15
Contents......Page 16
I Introduction to stochastic analysis......Page 20
1.1 Essentials of measure and probability......Page 22
1.2 Characteristic functions......Page 32
1.3 Conditioning......Page 35
1.4 Infinitely divisible and stable distributions......Page 40
1.5 Stable laws as the Holtzmark distributions......Page 46
1.6 Unimodality of probability laws......Page 48
1.7 Compactness for function spaces and measures......Page 54
1.8 Fractional derivatives and pseudo-differential operators......Page 61
1.9 Propagators and semigroups......Page 67
2.1 Random processes: basic notions......Page 77
2.2 Definition and basic properties of BM......Page 81
2.3 Construction via broken-line approximation......Page 85
2.4 Construction via Hilbert-space methods......Page 88
2.5 Construction via Kolmogorov’s continuity......Page 90
2.6 Construction via random walks and tightness......Page 92
2.7 Simplest applications of martingales......Page 95
2.8 Skorohod embedding and the invariance principle......Page 97
2.9 More advanced Hilbert space methods: Wiener chaos and stochastic integral......Page 100
2.10 Fock spaces, Hermite polynomials and Malliavin calculus......Page 106
2.11 Stationarity: OU processes and Holtzmark fields......Page 110
3.1 Definition of Lévy processes......Page 113
3.2 Poisson processes and integrals......Page 115
3.3 Construction of Lévy processes......Page 122
3.4 Subordinators......Page 127
3.5 Markov processes, semigroups and propagators......Page 129
3.6 Feller processes and conditionally positive operators......Page 134
3.7 Diffusions and jump-type Markov processes......Page 144
3.8 Markov processes on quotient spaces and reflections......Page 149
3.9 Martingales......Page 151
3.10 Stopping times and optional sampling......Page 157
3.11 Strong Markov property; diffusions as Feller processes with continuous paths......Page 162
3.12 Reflection principle and passage times......Page 166
4.1 Markov semigroups and evolution equations......Page 171
4.2 The Dirichlet problem for diffusion operators......Page 177
4.3 The stationary Feynman–Kac formula......Page 181
4.4 Diffusions with variable drift, Ornstein–Uhlenbeck processes......Page 184
4.5 Stochastic integrals and SDE based on Lévy processes......Page 186
4.6 Markov property and regularity of solutions......Page 191
4.7 Stochastic integrals and quadratic variation for square-integrable martingales......Page 197
4.8 Convergence of processes and semigroups......Page 206
4.9 Weak convergence of martingales......Page 212
4.10 Martingale problems and Markov processes......Page 214
4.11 Stopping and localization......Page 218
II Markov processes and beyond......Page 222
5 Processes in Euclidean spaces......Page 223
5.1 Direct analysis of regularity and well-posedness......Page 224
5.3 The Lie–Trotter type limits and T -products......Page 232
5.4 Martingale problems for Lévy type generators: existence......Page 240
5.5 Martingale problems for Lévy type generators: moments......Page 245
5.6 Martingale problems for Lévy type generators: unbounded coefficients......Page 247
5.7 Decomposable generators......Page 250
5.8 SDEs driven by nonlinear Lévy noise......Page 259
5.9 Stochastic monotonicity and duality......Page 269
5.10 Stochastic scattering......Page 274
5.11 Nonlinear Markov chains, interacting particles and deterministic processes......Page 276
5.12 Comments......Page 281
6.1 Stopped processes and boundary points......Page 289
6.2 Dirichlet problem and mixed initial-boundary problem......Page 293
6.3 The method of Lyapunov functions......Page 299
6.4 Local criteria for boundary points......Page 301
6.5 Decomposable generators in RdC......Page 305
6.6 Gluing boundary......Page 309
6.7 Processes on the half-line......Page 311
6.8 Generators of reflected processes......Page 312
6.9 Application to interacting particles: stochastic LLN......Page 314
6.10 Application to evolutionary games......Page 323
6.11 Application to finances: barrier options, credit derivatives, etc.......Page 326
6.12 Comments......Page 327
7.1 One-dimensional stable laws: asymptotic expansions......Page 329
7.2 Stable laws: asymptotic expansions and identities......Page 333
7.3 Stable laws: bounds......Page 338
7.4 Stable laws: auxiliary convolution estimates......Page 342
7.5 Stable-like processes: heat kernel estimates......Page 347
7.6 Stable-like processes: Feller property......Page 354
7.7 Application to sample-path properties......Page 355
7.8 Application to stochastic control......Page 359
7.9 Application to Langevin equations driven by a stable noise......Page 364
7.10 Comments......Page 367
8.1 Convergence of Markov semigroups and processes......Page 370
8.2 Diffusive approximations for random walks and CLT......Page 373
8.3 Stable-like limits for position-dependent random walks......Page 374
8.4 Subordination by hitting times and generalized fractional evolutions......Page 380
8.5 Limit theorems for position dependent CTRW......Page 388
8.6 Comments......Page 390
9.1 Infinitely-divisible complex distributions and complex Markov chains......Page 392
9.2 Path integral and perturbation theory......Page 399
9.3 Extensions......Page 404
9.4 Regularization of the Schrödinger equation by complex time or mass, or continuous observation......Page 409
9.5 Singular and growing potentials, magnetic fields and curvilinear state spaces......Page 412
9.6 Fock-space representation......Page 417
9.7 Comments......Page 419
Bibliography......Page 422
Index......Page 444