Stochastic equations in infinite dimensions:
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
2014
|
| Edition: | Second edition |
| Series: | Encyclopedia of mathematics and its applications
152 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverzeichnis Seite 446 - 490 |
| Physical Description: | xviii, 493 Seiten |
| ISBN: | 9781107055841 |
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| 245 | 1 | 0 | |a Stochastic equations in infinite dimensions |c Giuseppe Da Prato, Scuola Normale Superiore, Pisa; Jerzy Zabczyk, Polish Academy of Sciences |
| 250 | |a Second edition | ||
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Record in the Search Index
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| adam_text | Titel: Stochastic equations in infinite dimensions
Autor: Da Prato, Giuseppe
Jahr: 2014
Contents
Preface page xiii
Introduction: motivating examples 1
0.1 Lifts of diffusion processes 1
0.2 Markovian lifting of stochastic delay equations 2
0.3 Zakaï s equation 3
0.4 Random motion of a string 4
0.5 Stochastic equation of the free field 6
0.6 Equation of stochastic quantization 6
0.7 Reaction-diffusion equation 8
0.8 An example arising in neurophysiology 9
0.9 Equation of population genetics 9
0.10 Musiela s equation of the bond market 10
PART ONE FOUNDATIONS 13
1 Random variables 15
1.1 Random variables and their integrals 15
1.2 Operator valued random variables 23
1.3 Conditional expectation and independence 26
2 Probability measures 29
2.1 General properties 29
2.2 Gaussian measures in Banach spaces 36
2.2.1 Fernique theorem 36
2.2.2 Reproducing kernels 39
2.2.3 White noise expansions 42
2.3 Probability measures on Hilbert spaces 46
2.3.1 Gaussian measures on Hilbert spaces 46
2.3.2 Feldman-Hajek theorem 50
vi
Contents
2.3.3 An application to a general Cameron-Martin
formula 59
2.3.4 The Bochner theorem 60
3 Stochastic processes 65
3.1 General concepts 65
3.2 Kolmogorov test 67
3.3 Processes with filtration 71
3.4 Martingales 73
3.5 Stopping times and Markov processes 77
3.6 Gaussian processes in Hilbert spaces 77
3.7 Stochastic processes as random variables 78
4 The stochastic integral 80
4.1 Wiener processes 80
4.1.1 Hilbert space valued Wiener processes 81
4.1.2 Generalized Wiener processes on a Hilbert space 84
4.1.3 Wiener processes in U = L2(@) 86
4.1.4 Spatially homogeneous Wiener processes 90
4.1.5 Complements on a Brownian sheet 94
4.2 Definition of the stochastic integral 95
4.2.1 Stochastic integral for generalized Wiener
processes 100
4.2.2 Approximations of stochastic integrals 102
4.3 Properties of the stochastic integral 103
4.4 The Itô formula 106
4.5 Stochastic Fubini theorem 110
4.6 Basic estimates 114
4.7 Remarks on generalization of the integral 117
PART TWO EXISTENCE AND UNIQUENESS 119
5 Linear equations with additive noise 121
5.1 Basic concepts 121
5.1.1 Concept of solutions 121
5.1.2 Stochastic convolution 123
5.2 Existence and uniqueness of weak solutions 125
5.3 Continuity of weak solutions 129
5.3.1 Factorization formula 129
5.4 Regularity of weak solutions in the analytic case 134
5.4.1 Basic regularity theorems 134
5.4.2 Regularity in the border case 139
5.5 Regularity of weak solutions in the space of
continuous functions 143
Contents
vii
5.5.1 The case when A is self-adjoint 143
5.5.2 The case of a skew-symmetric generator 149
5.5.3 Equations with spatially homogeneous noise 150
5.6 Existence of strong solutions 156
6 Linear equations with multiplicative noise 159
6.1 Strong, weak and mild solutions 159
6.1.1 The case when B is bounded 164
6.2 Stochastic convolution for contraction semigroups 166
6.3 Stochastic convolution for analytic semigroups 168
6.3.1 General results 168
6.3.2 Variational case 171
6.3.3 Self-adjoint case 172
6.4 Maximal regularity for stochastic convolutions in Lp spaces 173
6.4.1 Maximal regularity 173
6.5 Existence of mild solutions in the analytic case 176
6.5.1 Introduction 176
6.5.2 Existence of solutions in the analytic case 176
6.6 Existence of strong solutions 181
7 Existence and uniqueness for nonlinear equations 186
7.1 Equations with Lipschitz nonlinearities 186
7.1.1 The case of cylindrical Wiener processes 196
7.2 Nonlinear equations on Banach spaces: additive noise 200
7.2.1 Locally Lipschitz nonlinearities 200
7.2.2 Dissipative nonlinearities 204
7.2.3 Dissipative nonlinearities by Euler approximations 207
7.2.4 Dissipative nonlinearities and general initial
conditions 210
7.2.5 Dissipative nonlinearities and general noise 213
7.3 Nonlinear equations on Banach spaces: multiplicative noise 215
7.4 Strong solutions 218
8 Martingale solutions 220
8.1 Introduction 220
8.2 Representation theorem 222
8.3 Compactness results 226
8.4 Proof of the main theorem 229
PART THREE PROPERTIES OF SOLUTIONS 233
9 Markov property and Kolmogorov equation 235
9.1 Regular dependence of solutions on initial data 235
9.1.1 Dilferentiability with respect to the initial condition 238
viii
Contents
9.1.2 Comments on stochastic flows 245
9.2 Markov and strong Markov properties 247
9.2.1 Case of Lipschitz nonlinearities 247
9.2.2 Markov property for equations in Banach spaces 252
9.3 Kolmogorov s equation: smooth initial functions 253
9.3.1 Bounded generators 254
9.3.2 Arbitrary generators 256
9.4 Further regularity properties of the transition semigroup 259
9.4.1 Linear case 259
9.4.2 Nonlinear case 266
9.5 Mild Kolmogorov equation 271
9.5.1 Solution of (9.75) 272
9.5.2 Identification of v(t, •) with P, p 21A
9.6 Specific examples 278
10 Absolute continuity and the Girsanov theorem 282
10.1 Absolute continuity for linear systems 282
10.1.1 The case B = B = / 287
10.2 Girsanov s theorem and absolute continuity for nonlinear
systems 291
10.2.1 Girsanov s theorem 291
10.3 Application to weak solutions 296
11 Large time behavior of solutions 300
11.1 Basic concepts 300
11.2 The Krylov-Bogoliubov existence theorem 304
11.2.1 Mixing and recurrence 307
11.2.2 Regular, strong Feller and irreducible semigroups 307
11.3 Linear equations with additive noise 308
11.3.1 Characterization theorem 310
11.3.2 Uniqueness of the invariant measure and asympotic
behavior 313
11.3.3 Strong Feller case 314
11.4 Linear equations with multiplicative noise 317
11.4.1 Bounded diffusion operators 317
11.4.2 Unbounded diffusion operators 322
11.5 General linear equations 324
11.6 Dissipative systems 326
11.6.1 Regular coefficients 327
11.6.2 Discontinuous coefficients 328
11.7 The compact case 332
11.7.1 Finite trace Wiener processes 333
11.7.2 Cylindrical Wiener processes 336
Contents
ix
12 Small noise asymptotic behavior 339
12.1 Large deviation principle 339
12.1.1 Formulation and basic properties 341
12.1.2 Lower estimates 341
12.1.3 Upper estimates 342
12.1.4 Change of variables 343
12.2 LDP for a family of Gaussian measures 344
12.3 LDP for Ornstein-Uhlenbeck processes 347
12.4 LDP for semilinear equations 350
12.5 Exit problem 351
12.5.1 Exit rate estimates 353
12.5.2 Exit place determination 358
12.5.3 Explicit formulae for gradient systems 363
13 Survey of specific equations 368
13.1 Countable systems of stochastic differential equations 368
13.2 Delay equations 369
13.3 First order equations 369
13.4 Reaction-diffusion equations 370
13.4.1 Spatially homogeneous noise 370
13.4.2 Skorohod equations in infinite dimensions 371
13.5 Equations for manifold valued processes 372
13.6 Equations with random boundary conditions 372
13.7 Equation of stochastic quantization 373
13.8 Filtering equations 375
13.9 Burgers equations 375
13.10 Kardar, Parisi and Zhang equation 376
13.11 Navier-Stokes equations and hydrodynamics 377
13.11.1 Existence and uniqueness for d = 2 377
13.11.2 Existence and uniqueness for d = 3 378
13.11.3 Stochastic magneto-hydrodynamics equations 379
13.11.4 The tamed Navier-Stokes equation 380
13.11.5 Renormalization of the Navier-Stokes equation 380
13.11.6 Euler equations 380
13.12 Stochastic climate models 380
13.13 Quasi-geostrophic equation 381
13.14 A growth of surface equation 381
13.15 Geometric SPDEs 382
13.16 Kuramoto-Sivashinsky equation 382
13.17 Cahn-Hilliard equations 383
13.18 Porous media equations 384
13.19 Korteweg-de Vries equation 386
X
Contents
13.19.1 Existence and uniqueness 386
13.19.2 Soliton dynamic 386
13.20 Stochastic conservation laws 386
13.21 Wave equations 387
13.21.1 Spatially homogeneous noise 388
13.21.2 Symmetric hyperbolic systems 389
13.21.3 Wave equations in Riemannian manifolds 389
13.22 Beam equations 389
13.23 Nonlinear Schrödinger equations 390
13.23.1 Existence and uniqueness 390
13.23.2 Blow-up 391
14 Some recent developments 392
14.1 Complements on solutions of equations 392
14.1.1 Stochastic PDEs in Banach spaces 3 92
14.1.2 Backward stochastic differential equations 393
14.1.3 Wiener chaos expansions 395
14.1.4 Hida s white noise approach 395
14.1.5 Rough paths approach 396
14.1.6 Equations with fractional Brownian motion 398
14.1.7 Equations with Lévy noise 398
14.1.8 Equations with irregular coefficients 399
14.1.9 Yamada-Watanabe theory in infinite dimensions 399
14.1.10 Numerical methods for SPDEs 399
14.2 Some results on laws of solutions 400
14.2.1 Applications of Malliavin calculus 400
14.2.2 Fokker-Planck and mass transport equations 401
14.2.3 Ultraboundedness and Harnack inequalities 402
14.2.4 Gradient flows in Wasserstein spaces and Dirichlet
forms 402
14.3 Asymptotic properties of the solutions 403
14.3.1 More on invariant measures 403
14.3.2 More on large deviations 404
14.3.3 Stochastic resonance 404
14.3.4 Averaging 404
14.3.5 Short time asymptotic 405
Appendix A Linear deterministic equations 406
A.l Cauchy problems and semigroups 406
A.2 Basic properties of Co-semigroups 407
A.3 Cauchy problem for nonhomogeneous equations 409
A.4 Cauchy problem for analytic semigroups 412
A. 5 Example of deterministic systems 419
Contents xi
AppendixB Some results on control theory 428
B. 1 Controllability and stabilizability 428
B.2 Comparison of images of linear operators 429
B.3 Operators associated with control systems 431
AppendixC Nuclear and Hilbert-Schmidt operators 436
AppendixD Dissipative mappings 440
D. 1 Subdifferential of the norm 440
D.2 Dissipative mappings 442
D.3 Continuous dissipative mappings 444
Bibliography 446
Index 491
|
| any_adam_object | 1 |
| author | Da Prato, Giuseppe 1936-2023 Zabczyk, Jerzy 1941- |
| author_GND | (DE-588)121352641 (DE-588)12135234X |
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| author_sort | Da Prato, Giuseppe 1936-2023 |
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| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2 |
| dewey-search | 519.2 |
| dewey-sort | 3519.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Second edition |
| format | Book |
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| isbn | 9781107055841 |
| language | English |
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| publisher | Cambridge University Press |
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| series | Encyclopedia of mathematics and its applications |
| series2 | Encyclopedia of mathematics and its applications |
| spelling | Da Prato, Giuseppe 1936-2023 Verfasser (DE-588)121352641 aut Stochastic equations in infinite dimensions Giuseppe Da Prato, Scuola Normale Superiore, Pisa; Jerzy Zabczyk, Polish Academy of Sciences Second edition Cambridge Cambridge University Press 2014 xviii, 493 Seiten txt rdacontent n rdamedia nc rdacarrier Encyclopedia of mathematics and its applications 152 Literaturverzeichnis Seite 446 - 490 Stochastische Differentialgleichung (DE-588)4057621-8 gnd rswk-swf Unendlichdimensionaler Raum (DE-588)4207852-0 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 s Unendlichdimensionaler Raum (DE-588)4207852-0 s DE-604 Zabczyk, Jerzy 1941- Verfasser (DE-588)12135234X aut Encyclopedia of mathematics and its applications 152 (DE-604)BV000903719 152 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027079620&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Da Prato, Giuseppe 1936-2023 Zabczyk, Jerzy 1941- Stochastic equations in infinite dimensions Encyclopedia of mathematics and its applications Stochastische Differentialgleichung (DE-588)4057621-8 gnd Unendlichdimensionaler Raum (DE-588)4207852-0 gnd |
| subject_GND | (DE-588)4057621-8 (DE-588)4207852-0 |
| title | Stochastic equations in infinite dimensions |
| title_auth | Stochastic equations in infinite dimensions |
| title_exact_search | Stochastic equations in infinite dimensions |
| title_full | Stochastic equations in infinite dimensions Giuseppe Da Prato, Scuola Normale Superiore, Pisa; Jerzy Zabczyk, Polish Academy of Sciences |
| title_fullStr | Stochastic equations in infinite dimensions Giuseppe Da Prato, Scuola Normale Superiore, Pisa; Jerzy Zabczyk, Polish Academy of Sciences |
| title_full_unstemmed | Stochastic equations in infinite dimensions Giuseppe Da Prato, Scuola Normale Superiore, Pisa; Jerzy Zabczyk, Polish Academy of Sciences |
| title_short | Stochastic equations in infinite dimensions |
| title_sort | stochastic equations in infinite dimensions |
| topic | Stochastische Differentialgleichung (DE-588)4057621-8 gnd Unendlichdimensionaler Raum (DE-588)4207852-0 gnd |
| topic_facet | Stochastische Differentialgleichung Unendlichdimensionaler Raum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027079620&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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