Stochastic partial differential equations and applications

Contents\r......Page 4
1. Introduction......Page 9
2. Matrix elements of the SWN operators......Page 10
3. Commutation relations of the SWN operators......Page 15
4. Inclusion of the SWN calculus in the representation free quantum stochastic calculus of Accardi, Fagnola, and Quaegebeur......Page 19
5. On the connection between the Finite Difference operators and the creation, annihilation, and conservation operators......Page 24
REFERENCES......Page 26
1 Introduction......Page 28
2 Construction of Euclidean random fields via (modified) covariant SPDEs......Page 30
3 Analytic continuation of the Schwinger functions......Page 35
4 Quantum fields with indefinite metric......Page 37
5 On the construction of asymptotic states and the S\'-matrix......Page 39
References......Page 42
1 Introduction......Page 46
2 Setting of the problem in a general framework......Page 47
3 Riccati Characterization......Page 50
4 Duality Relation......Page 51
5 Carleman Estimates......Page 54
6 Robust solutions of backward equations......Page 56
References......Page 57
1 Introduction......Page 59
2 Linear SDE\'s and instruments......Page 60
3 Nonlinear SDE\'s and a posteriori states......Page 64
4 Invariant measures......Page 66
References......Page 70
1. INTRODUCTION......Page 74
2. ABSOLUTE CONTINUITY AND SOBOLEV REGULARITY OP SOLUTIONS......Page 76
3. EXISTENCE AND UNIQUENESS OF SOLUTIONS AND ASSOCIATED SEMIGROUPS......Page 78
4. INFINITE DIMENSIONAL CASE......Page 83
REFERENCES......Page 90
1 Introduction......Page 93
2 Deterministic theory of attractors......Page 94
3 The stochastic case: difficulties......Page 95
4 Concepts of attraction: a comparison study......Page 97
5 Asymptotic behaviour from random attractors......Page 100
6 Structure of attractors and some other problems......Page 102
References......Page 104
0. Introduction......Page 137
1. Preliminaries — martingale solutions......Page 140
3. Properties of (Pt) — the case of bounded F......Page 142
4. The semigroup (Pt) — the case of general F......Page 146
5. Examples......Page 147
REFERENCES......Page 150
1. INTRODUCTION......Page 153
2. PRELIMINARIES......Page 154
3. GENERALIZED SOLUTIONS OF FIRST-ORDER SPDEs......Page 156
4. REGULARITY OF GENERALIZED SOLUTIONS......Page 158
REFERENCES......Page 163
1. INTRODUCTION......Page 165
2. DIFFERENTIABILITY OF THE ITO PARALLEL TRANSPORT AND INTERTWINING FORMULA......Page 172
3. THE SPACE OF TANGENT PROCESSES......Page 177
4. INTEGRATION BY PARTS ON THE PATH SPACE......Page 179
5. STRUCTURAL EQUATIONS OF THE PATH SPACE......Page 180
6. RlEMANNIAN CONNECTIONS......Page 182
7. WEITZENBOCK FORMULAE......Page 184
8. ANTICIPATIVE INTEGRALS AND WEITZENBOCK FORMULAE......Page 190
9. ADAPTED DIFFERENTIAL GEOMETRY......Page 192
REFERENCES......Page 195
1 Introduction......Page 198
2 Regularity properties of Pt(p......Page 200
3 A proof of hypercontractivity......Page 206
4 Appendix......Page 208
References......Page 212
1. INTRODUCTION......Page 214
2. WHITE NOISE ANALYSIS......Page 216
3. APPLICATION OF CONTRACTION METHODS......Page 219
4. APPLICATION OF STOCHASTIC REPRESENTATION FORMULAE......Page 223
REFERENCES......Page 225
1. INTRODUCTION......Page 227
2. THE MINIMAL QMS......Page 230
3. ASYMPTOTIC BEHAVIOUR......Page 231
4. INVARIANT STATES......Page 232
5. FAITHFULNESS AND IRREDUCIBILITY......Page 234
6. APPROACH TO EQUILIBRIUM......Page 236
REFERENCES......Page 237
1 Introduction......Page 239
2.1 Uniqueness of weak solutions......Page 240
2.2 Fluctuations and singularities......Page 241
2.3 Example: uniqueness for one-dimensional ODEs......Page 242
2.3.2 Proof of the theorem......Page 243
3 Singularities of stationary and transient flows......Page 245
3.1 The result for stationary solutions......Page 246
4 Probabilistic description of vortex filaments......Page 249
4.1 Cross-section......Page 252
4.2 Interaction energy and first results......Page 254
4.3 Spectral representation and Gibbs measures......Page 257
References......Page 259
1.1. Interfaces and effective models......Page 265
1.2. A massless dynamical model: the Ginzburg— Landau Ve interface model......Page 266
1.3. Gibbsian invariant measures......Page 267
1.4. A random field of increments......Page 268
1.5. A probabilistic analysis of random interfaces......Page 269
2.1. The Punaki-Spohn classification result......Page 270
2.3. The HS representation in the massless context......Page 271
2.5. A space— time correlation formula......Page 273
3.2. Motion by mean curvature......Page 274
4.1. A Large Deviation principle......Page 276
4.3. Hydrodynamic limit and large deviations......Page 277
5.1. The fluctuation field and its scaling limit......Page 278
5.2. Homogenization for a random walk in random environment......Page 279
6.2. Some open problems......Page 280
REFERENCES......Page 281
1 Introduction......Page 284
2 Description of a model problem......Page 288
3 Existence and uniqueness results for HJB equations......Page 291
4 Approximation of mild solutions......Page 296
5 Application to stochastic control problems: verification theorems and optimal feedbacks......Page 299
6.1 An example with distributed control, quadratic Hamiltonian......Page 301
6.2 An example of boundary control problem......Page 302
7 Some history......Page 304
References......Page 308
1. INTRODUCTION......Page 315
2. EXISTENCE AND UNIQUENESS OF THE SOLUTION......Page 316
3. WONG-ZAKAI TYPE APPROXIMATIONS......Page 318
4. SPLITTING UP METHOD......Page 321
5. FINITE DIFFERENCE APPROXIMATIONS......Page 325
6. APPLICATIONS TO NONLINEAR FILTERING......Page 328
REFERENCES......Page 332
1 Introduction......Page 336
2 Stochastic heat and wave equations......Page 337
3 Stochastic Volterra equation......Page 342
References......Page 347
1. INTRODUCTION......Page 351
2. THE CAUCHY PROBLEM......Page 353
3. EQUATIONS IN DOMAINS......Page 356
REFERENCES......Page 360
1. INTRODUCTION......Page 363
(A). Strong solutions and mild solutions......Page 364
(B). Some stability definitions......Page 366
(A). Linear deterministic equations......Page 368
(B). Linear stochastic evolution equations......Page 371
(A) . Equivalence of p-th moment integrability and exponential stability......Page 374
(B). Stability criteria based on a coercivity condition......Page 375
(C). Lyapunov functions and stability......Page 378
(D). Stabilization of stochastic (deterministic) evolution equations......Page 381
4. STABILITY AND SOME RELATED TOPICS......Page 383
(A). Stability with a certain decay rate......Page 384
(B). Stability of stochastic delay evolution equations......Page 386
(C). Ultimate boundedness and invariant measures......Page 388
(D). Some related topics......Page 391
REFERENCES......Page 392
0. Introduction......Page 398
1. Basic definitions and general results......Page 399
2. Analytic approach......Page 403
3. Probabilistic approach......Page 406
References......Page 409
1 Introduction......Page 413
2 Stochastic 2-D Navier-Stokes Equation......Page 415
3 Markov-Feller Process......Page 419
4 Stopping Time Problem......Page 422
5 Impulse Control Problem......Page 423
References......Page 426
1 Introduction......Page 429
2.2 The Navier-Stokes equation......Page 431
2.3 Approximations of Navier-Stokes equation......Page 432
2.3.1 Linearization of the quadratic term......Page 433
2.3.2 Construction of an approximating sequence......Page 434
2.3.3 Weak compactness of approximations......Page 435
2.4 Existence of weak global solutions......Page 437
References......Page 439
0. INTRODUCTION......Page 440
0.1. Parabolic equation......Page 441
0.2. Wave equation......Page 443
1.2. Spatially homogeneous Wiener......Page 444
1.2 Stochastic integration in weighted I/p-spaces......Page 446
2. DEFINITIONS OF SOLUTIONS TO STOCHASTIC PARABOLIC AND WAVE EQUATIONS......Page 447
3. EXISTENCE OP SOLUTION......Page 448
REFERENCES......Page 449
2. PRELIMINARIES ON MALLIAVIN CALCULUS......Page 451
3. LAWS OF SOLUTIONS OF SPDE\'S......Page 453
3.1 Existence and smoothness of density......Page 455
3.1.1 Examples in the one-dimensional case......Page 457
3.1.3 Remarks......Page 458
3.2 Further properties of the density......Page 459
3.3 Nonlinear operators......Page 460
References......Page 461
Stochastic Curvature Driven Flows......Page 465
1 Motion by Mean Curvature......Page 466
1.1 Methods of Solution......Page 467
2 Stochastic Curvature Driven Flows......Page 469
2.1 Stochastic Motion by Mean Curvature......Page 470
2.2 Stochastic Dendritic Crystal Growth......Page 472
2.3 Stochastic Viscosity Solution......Page 473
3 Noise and Uniqueness of Motion by Mean Curvature......Page 474
4 Open Problems......Page 476
References......Page 477