Stochastic Equations in Infinite Dimensions

Cover......Page 1
Half title......Page 3
Series......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 15
 0.1 Lifts of diffusion processes......Page 21
 0.2 Markovian lifting of stochastic delay equations......Page 22
 0.3 Zakaï\'s equation......Page 23
 0.4 Random motion of a string......Page 24
 0.6 Equation of stochastic quantization......Page 26
 0.7 Reaction-diffusion equation......Page 28
 0.9 Equation of population genetics......Page 29
 0.10 Musiela\'s equation of the bond market......Page 30
Part I Foundations......Page 33
  1.1 Random variables and their integrals......Page 35
  1.2 Operator valued random variables......Page 43
  1.3 Conditional expectation and independence......Page 46
  2.1 General properties......Page 49
   2.2.1 Fernique theorem......Page 56
   2.2.2 Reproducing kernels......Page 59
   2.2.3 White noise expansions......Page 62
   2.3.1 Gaussian measures on Hilbert spaces......Page 66
   2.3.2 Feldman–Hajek theorem......Page 70
   2.3.3 An application to a general Cameron–Martin formula......Page 79
   2.3.4 The Bochner theorem......Page 80
  3.1 General concepts......Page 85
  3.2 Kolmogorov test......Page 87
  3.3 Processes with filtration......Page 91
  3.4 Martingales......Page 93
  3.6 Gaussian processes in Hilbert spaces......Page 97
  3.7 Stochastic processes as random variables......Page 98
  4.1 Wiener processes......Page 100
   4.1.1 Hilbert space valued Wiener processes......Page 101
   4.1.2 Generalized Wiener processes on a Hilbert space......Page 104
   4.1.3 Wiener processes in U = L[sup(2)](O)......Page 106
   4.1.4 Spatially homogeneous Wiener processes......Page 110
   4.1.5 Complements on a Brownian sheet......Page 114
  4.2 Definition of the stochastic integral......Page 115
   4.2.1 Stochastic integral for generalized Wiener processes......Page 120
   4.2.2 Approximations of stochastic integrals......Page 122
  4.3 Properties of the stochastic integral......Page 123
  4.4 The Ito formula......Page 126
  4.5 Stochastic Fubini theorem......Page 130
  4.6 Basic estimates......Page 134
  4.7 Remarks on generalization of the integral......Page 137
Part II Existence and uniqueness......Page 139
   5.1.1 Concept of solutions......Page 141
   5.1.2 Stochastic convolution......Page 143
  5.2 Existence and uniqueness of weak solutions......Page 145
   5.3.1 Factorization formula......Page 149
   5.4.1 Basic regularity theorems......Page 154
   5.4.2 Regularity in the border case......Page 159
   5.5.1 The case when A is self-adjoint......Page 163
   5.5.2 The case of a skew-symmetric generator......Page 169
   5.5.3 Equations with spatially homogeneous noise......Page 170
  5.6 Existence of strong solutions......Page 176
  6.1 Strong, weak and mild solutions......Page 179
   6.1.1 The case when B is bounded......Page 184
  6.2 Stochastic convolution for contraction semigroups......Page 186
   6.3.1 General results......Page 188
   6.3.2 Variational case......Page 191
   6.3.3 Self-adjoint case......Page 192
   6.4.1 Maximal regularity......Page 193
   6.5.2 Existence of solutions in the analytic case......Page 196
  6.6 Existence of strong solutions......Page 201
  7.1 Equations with Lipschitz nonlinearities......Page 206
   7.1.1 The case of cylindrical Wiener processes......Page 216
   7.2.1 Locally Lipschitz nonlinearities......Page 220
   7.2.2 Dissipative nonlinearities......Page 224
   7.2.3 Dissipative nonlinearities by Euler approximations......Page 227
   7.2.4 Dissipative nonlinearities and general initial conditions......Page 230
   7.2.5 Dissipative nonlinearities and general noise......Page 233
  7.3 Nonlinear equations on Banach spaces: multiplicative noise......Page 235
  7.4 Strong solutions......Page 238
  8.1 Introduction......Page 240
  8.2 Representation theorem......Page 242
  8.3 Compactness results......Page 246
  8.4 Proof of the main theorem......Page 249
Part III Properties of solutions......Page 253
  9.1 Regular dependence of solutions on initial data......Page 255
   9.1.1 Differentiability with respect to the initial condition......Page 258
   9.1.2 Comments on stochastic flows......Page 265
   9.2.1 Case of Lipschitz nonlinearities......Page 267
   9.2.2 Markov property for equations in Banach spaces......Page 272
  9.3 Kolmogorov\'s equation: smooth initial functions......Page 273
   9.3.1 Bounded generators......Page 274
   9.3.2 Arbitrary generators......Page 276
   9.4.1 Linear case......Page 279
   9.4.2 Nonlinear case......Page 286
  9.5 Mild Kolmogorov equation......Page 291
   9.5.1 Solution of (9.75)......Page 292
   9.5.2 Identification of v(t, ・) with P[sub(t)]φ......Page 294
  9.6 Specific examples......Page 298
  10.1 Absolute continuity for linear systems......Page 302
   10.1.1 The case B = B = I......Page 307
   10.2.1 Girsanov\'s theorem......Page 311
  10.3 Application to weak solutions......Page 316
  11.1 Basic concepts......Page 320
  11.2 The Krylov–Bogoliubov existence theorem......Page 324
   11.2.2 Regular, strong Feller and irreducible semigroups......Page 327
  11.3 Linear equations with additive noise......Page 328
   11.3.1 Characterization theorem......Page 330
   11.3.2 Uniqueness of the invariant measure and asympotic behavior......Page 333
   11.3.3 Strong Feller case......Page 334
   11.4.1 Bounded diffusion operators......Page 337
   11.4.2 Unbounded diffusion operators......Page 342
  11.5 General linear equations......Page 344
  11.6 Dissipative systems......Page 346
   11.6.1 Regular coefficients......Page 347
   11.6.2 Discontinuous coefficients......Page 348
  11.7 The compact case......Page 352
   11.7.1 Finite trace Wiener processes......Page 353
   11.7.2 Cylindrical Wiener processes......Page 356
  12.1 Large deviation principle......Page 359
   12.1.2 Lower estimates......Page 361
   12.1.3 Upper estimates......Page 362
   12.1.4 Change of variables......Page 363
  12.2 LDP for a family of Gaussian measures......Page 364
  12.3 LDP for Ornstein--Uhlenbeck processes......Page 367
  12.4 LDP for semilinear equations......Page 370
  12.5 Exit problem......Page 371
   12.5.1 Exit rate estimates......Page 373
   12.5.2 Exit place determination......Page 378
   12.5.3 Explicit formulae for gradient systems......Page 383
  13.1 Countable systems of stochastic differential equations......Page 388
  13.3 First order equations......Page 389
   13.4.1 Spatially homogeneous noise......Page 390
   13.4.2 Skorohod equations in infinite dimensions......Page 391
  13.6 Equations with random boundary conditions......Page 392
  13.7 Equation of stochastic quantization......Page 393
  13.9 Burgers equations......Page 395
  13.10 Kardar, Parisi and Zhang equation......Page 396
   13.11.1 Existence and uniqueness for d = 2......Page 397
   13.11.2 Existence and uniqueness for d = 3......Page 398
   13.11.3 Stochastic magneto-hydrodynamics equations......Page 399
  13.12 Stochastic climate models......Page 400
  13.14 A growth of surface equation......Page 401
  13.16 Kuramoto–Sivashinsky equation......Page 402
  13.17 Cahn–Hilliard equations......Page 403
  13.18 Porous media equations......Page 404
  13.20 Stochastic conservation laws......Page 406
  13.21 Wave equations......Page 407
   13.21.1 Spatially homogeneous noise......Page 408
  13.22 Beam equations......Page 409
   13.23.1 Existence and uniqueness......Page 410
   13.23.2 Blow-up......Page 411
   14.1.1 Stochastic PDEs in Banach spaces......Page 412
   14.1.2 Backward stochastic differential equations......Page 413
   14.1.4 Hida\'s white noise approach......Page 415
   14.1.5 Rough paths approach......Page 416
   14.1.7 Equations with Lévy noise......Page 418
   14.1.10 Numerical methods for SPDEs......Page 419
   14.2.1 Applications of Malliavin calculus......Page 420
   14.2.2 Fokker–Planck and mass transport equations......Page 421
   14.2.4 Gradient flows in Wasserstein spaces and Dirichlet forms......Page 422
   14.3.1 More on invariant measures......Page 423
   14.3.4 Averaging......Page 424
   14.3.5 Short time asymptotic......Page 425
 A.1 Cauchy problems and semigroups......Page 426
 A.2 Basic properties of C[sup(0)]-semigroups......Page 427
 A.3 Cauchy problem for nonhomogeneous equations......Page 429
  A.4.1 Analytic generators......Page 432
  A.4.2 Variational generators......Page 433
  A.4.3 Fractional powers and interpolation spaces......Page 434
  A.4.4 Regularity of solutions for the homogeneous Cauchy problem......Page 437
  A.4.5 Regularity for the nonhomogeneous problem......Page 438
  A.5.1 Delay systems......Page 439
  A.5.2 Heat equation......Page 440
  A.5.3 Heat equation in variational form......Page 442
  A.5.4 Wave and plate equations......Page 444
  A.5.5 Wave and plate equation with strong damping......Page 445
 B.1 Controllability and stabilizability......Page 448
 B.2 Comparison of images of linear operators......Page 449
 B.3 Operators associated with control systems......Page 451
  B.3.1 Characterization of L(U)......Page 452
  B.3.2 Characterization of the image of L......Page 453
 Appendix C: Nuclear and Hilbert–Schmidt operators......Page 456
 D.1 Subdifferential of the norm......Page 460
 D.2 Dissipative mappings......Page 462
 D.3 Continuous dissipative mappings......Page 464
Bibliography......Page 466
Index......Page 511