Trends in Stochastic Analysis

Title......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 8
Heinrich von Weizsäcker’s students
......Page 10
Heinrich von Weizsäcker’s publications
......Page 12
I. Foundations and techniques in stochastic analysis......Page 18
1.1 Introduction......Page 20
1.2 Events and random variables – an outline......Page 24
1.3 Spaces with denumerable separation......Page 26
1.4 The axioms for random variables......Page 29
1.5 Events......Page 32
1.6 Equality and a.s. equality......Page 36
Bibliography......Page 41
2.1 Introduction......Page 42
2.2 The competitors......Page 44
2.3 Chaining at work......Page 47
2.4 Examples and complements......Page 53
2.5 Dispersion of sets: upper bounds......Page 59
2.6 Appendix: Proofs of the chaining lemmas......Page 64
Bibliography......Page 69
3.1 Introduction......Page 72
3.1.1 Notation......Page 74
3.2 Skew-products......Page 75
3.2.1 Admissible measures......Page 77
3.2.2 A simple example......Page 78
3.3 Skew-products of Markov processes versus random dynamical systems......Page 79
3.4 Ergodicity criteria for Markov semigroups......Page 82
3.4.1 Off-white noise systems......Page 85
3.4.2 Another quasi-Markov property......Page 87
3.5 The Gaussian case......Page 89
3.5.1 The quasi-Markov property......Page 91
3.5.2 The strong Feller property......Page 93
3.5.3 The off-white noise case......Page 97
3.6 Appendix A: Equivalence of the strong and ultra Feller properties......Page 98
3.7 Appendix B: Some Gaussian measure theory......Page 101
Bibliography......Page 103
4.1 Introduction......Page 106
4.2 The speed of fragmentation in stick-breaking......Page 108
4.3 A polymer model in a random environment......Page 114
4.4 The speed of emergence in Kingman’s coalescent......Page 121
4.5 Conclusion......Page 126
Bibliography......Page 127
II. Construction, simulation, discretization of stochastic processes......Page 128
Abstract......Page 130
5.1 Introduction......Page 131
5.2 Example: confinement to a curved planar wire......Page 133
5.2.1 Geometry: an embedded curved wire......Page 135
5.2.2 Rescaling......Page 136
5.2.3 Renormalization......Page 137
5.2.4 Epiconvergence......Page 143
5.2.5 Strong convergence of the generators......Page 146
5.2.6 The result for the induced metric......Page 149
5.3 Conditioned Brownian motion......Page 151
5.4 The case of Riemannian submanifolds......Page 153
5.5 Two limits......Page 156
5.6 Two open problems......Page 162
5.6.1 Tubes with fibres of variable shape......Page 163
Bibliography......Page 164
6.1 Introduction......Page 166
6.2.1 The Metropolis–Hastings algorithm......Page 168
6.2.2 Langevin sampling......Page 170
6.3.1 Linear equations......Page 171
6.3.2 Semilinear equations......Page 173
6.4 Conditioned diffusions......Page 177
6.5.1 Construction of the smoothing SPDE......Page 182
6.5.2 Some remarks about smoothing......Page 187
6.6 Metropolis–Hastings algorithm on path space......Page 188
6.7 Conclusion......Page 191
Bibliography......Page 192
7.1 Introduction......Page 194
7.1.1 The information constraints......Page 197
7.1.2 Synopsis......Page 199
7.2.1 Classical setting......Page 200
7.2.2 Orlicz-norm distortion......Page 204
7.3 Gaussian signals......Page 208
7.3.1 Asymptotic estimates......Page 209
7.3.2 Some examples......Page 210
7.3.3 A particular random coding strategy......Page 211
7.3.4 The fractional Brownian motion......Page 212
7.4.1 The mutual information and Shannon’s Source Coding Theorem......Page 214
7.4.2 The derivation of the distortion rate function D(r, 2)......Page 216
7.4.3 Generalizations to other coding quantities (first approach)......Page 220
7.4.4 Generalizations to other coding quantities (second approach)......Page 221
7.4.5 A particular random quantization procedure......Page 223
7.5 Diffusions......Page 225
7.5.1 The decoupling method......Page 227
7.5.2 The corresponding rate allocation problem......Page 230
7.6.1 Estimates based on moment conditions on the increments......Page 233
Bibliography......Page 236
III. Stochastic analysis in mathematical physics......Page 240
Abstract......Page 242
8.1.1 Motivation......Page 243
8.1.2 Intermittency......Page 244
8.2.1 Model......Page 245
8.2.2 Main theorems......Page 246
8.2.3 Discussion......Page 248
8.3.1 Model......Page 249
8.3.2 Main theorems......Page 250
8.4 Symmetric voter model......Page 251
8.4.2 Main theorems......Page 252
8.4.3 Discussion......Page 253
Bibliography......Page 254
9.1 Introduction......Page 256
9.2 What is a stochastic dynamical system?......Page 257
9.3 Spectral theory of linear cocycles: hyperbolicity......Page 258
9.4 The local stable manifold theorem......Page 263
9.5 Stochastic systems with finite memory......Page 268
9.5.1 Existence of cocycles for regular sfde’s......Page 269
9.6 Semilinear see’s......Page 273
9.6.1 Smooth cocycles for semilinear see’s and spde’s......Page 274
9.7 Examples: semilinear spde’s......Page 280
9.8.1 Anticipating semilinear sfde’s......Page 282
9.8.2 Anticipating semilinear see’s......Page 286
Bibliography......Page 287
10.1 Introduction......Page 290
10.2 Feynman formulae, Feynman pseudo-measures and Feynman integrals......Page 292
10.3 Feynman formulae via the Chernoff theorem......Page 297
10.4 Representation of solutions to the Schr¨odinger equation by the Feynman integral over trajectories in phase space......Page 298
10.5 Feynman formulae for the Cauchy-Dirichlet problem [20]......Page 300
10.6 Stochastic Schr¨odinger–Ito equation (Belavkin equation) with two-dimensional white noise [9]......Page 301
10.7 Equations on manifolds of mappings ([21])......Page 302
10.8 Feynman formulae for the diffusion equation with coordinate dependent diffusion coefficient and for the Schrödinger equation with coordinate dependent mass......Page 306
Bibliography......Page 308
11.1 Introduction......Page 310
11.2 Wiener space and Fock space......Page 314
11.3 Quantization and the Malliavin Calculus......Page 319
11.4 Quantization and white noise analysis......Page 322
11.5 Fedosov quantization for Taubes limit model......Page 326
Bibliography......Page 330
IV. Stochastic analysis in mathematical biology......Page 334
12.1 Introduction......Page 336
12.2 Population genetic models with neutral types......Page 338
12.2.1 “Classical” limit results in the finite variance regime......Page 339
12.2.2 Beyond finite variance: occasional extremer eproduction events......Page 342
12.2.3 Introducing mutation......Page 345
12.2.4 Lookdown......Page 350
12.3 Neutral genealogies: beyond Kingman’s coalescent......Page 353
12.3.1 Genealogies and coalescent processes......Page 354
12.4 Population genetic inference......Page 359
12.4.1 Finite-alleles recursion I: Using the lookdown construction......Page 360
12.4.2 Finite-alleles recursion II: Generator approach......Page 362
12.4.3 A Monte Carlo scheme for sampling probabilities......Page 365
12.4.4 Simulating samples......Page 366
Bibliography......Page 368
Abstract......Page 372
13.1 Introduction......Page 373
13.2 The discrete ratchet – Haigh’s approach......Page 376
13.3 The Fleming–Viot diffusion......Page 379
13.4 The infinite population limit......Page 380
13.5 One-dimensional diffusion approximations......Page 384
13.6 Simulations......Page 390
13.7 Discussion......Page 394
Bibliography......Page 396