An introduction to infinite dimensional analysis:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Pisa
Scuola Normale Superiore
2001
|
| Schriftenreihe: | Appunti
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | VII, 184 S. |
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Datensatz im Suchindex
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|---|---|
| adam_text | Titel: An introduction to infinite dimensional analysis
Autor: Da Prato, Giuseppe
Jahr: 2001
Contents
1 Gaussian measures in Hilbert spaces 1
1.1 One dimensional Hilbert spaces..................................1
1.2 Finite dimensional Hilbert spaces................................2
1.2.1 Product probabilities......................................3
1.2.2 Definition of gaussian measures..........................3
1.3 Infinite dimensional Hilbert spaces ..............................4
1.3.1 Some preliminaries........................................5
1.3.2 Definition of gaussian measures..........................7
1.4 Gaussian random variables........................................14
1.4.1 Law of a random variable ................................14
1.4.2 Change of variables of gaussian measures........14
1.4.3 Independent gaussian random variables..................16
1.5 The reproducing kernel and the white noise function............17
2 The Cameron-Martin formula 21
2.1 Introduction and setting of the problem.............21
2.2 Equivalence and singularity of product measures........22
2.3 Proof of the Cameron-Martin formula .............27
2.4 The Feldman-Hajek theorem ..................28
3 Brownian motion 31
3.1 Construction of a Brownian motion...............31
3.2 Variation of a Brownian motion.................34
3.3 Wiener integral..........................37
3.4 Multidimensional Brownian motions...............40
3.5 Law of a Brownian motion in L2(0,1)..............42
3.5.1 Brownian Bridge.....................45
vj Contents
4 Stochastic perturbations of a dynamical system 47
4.1 Introduction.............................47
4.2 The Ornstein-Uhlenbeck process ................52
4.3 The transition semigroup in the deterministic case.......53
4.4 The transition semigroup in the stochastic case ........55
4.5 A generalization..........................52
5 Invariant measures for Markov semigroups • 65
5.1 Markov semigroups........................65
5.2 Invariant measures........................67
5.3 Ergodic means...........................71
5.4 The Von Neumann theorem...................73
5.5 Ergodicity.............................74
5.6 Properties of the set of all invariant measures .........77
6 Weak convergence of measures 61
6.1 Some additional properties of measures.............61
6.2 Positive functional........................84
6.3 The Prokhorov theorem.....................88
7 Existence and uniqueness of invariant measures 93
7.1 The Krylov-Bogoliubov theorem.................93
7.1.1 Application to differential stochastic equations in R . 97
7.2 Uniqueness of invariant measures................99
8 Examples of Markov semigroups 101
8.1 Introduction............................101
8.2 The Heat semigroup .......................102
8.3 The Ornstein-Uhlenbeck semigroup...............108
8.3.1 Regularizing property of the O. U. semigroup.....110
8.3.2 Invariant Measures ....................113
8.4 Application to differential stochastic equations.........114
9 L2 and Sobolev spaces with respect to a Gaussian measure 119
9.1 Notations.............................119
9.2 Orthonormal basis in L2(H,/i)..................120
9.2.1 The one-dimensional case................120
9.2.2 The infinite dimensional case .............. 122
Contents vii
9.3 Ito-Wiener decomposition....................125
9.4 The Ornstein-Uhlenbeck semigroup...............127
9.5 The Sobolev space W1,2(H,fi)..................129
9.5.1 The space L2(i7, //; H)..................132
9.6 The Dirichlet form associated to fi ...............133
9.6.1 Poincare and Log Sobolev inequalities..........136
9.7 Compactness of the embedding W1,2(H,fi) C L2(//,/x) .... 142
9.8 The Sobolev space IV2,2 (i/,//)..................145
10 Gibbs measures 149
10.1 Introduction and setting of the problem.............149
10.2 Gibbs measures on L2(0,1)....................151
10.3 The Sobolev space W1,2(H, v)..................153
10.4 Symmetricity of the operator N.................155
10.5 Some complements on differential stochastic equations .... 157
10.5.1 Cylindrical Wiener process and stochastic convolution . 157
10.5.2 Differential stochastic equations.............160
10.6 Self-adjointness of .......................162
A Linear Semigroups Theory 167
A.l Some preliminaries on spectral theory..............167
A.2 Strongly continuous semigroups.................169
A.3 The Hille-Yosida theorem....................173
A.3.1 Cores............................178
A.4 Dissipative operators.......................179
Bibliography 183
|
| adam_txt |
Titel: An introduction to infinite dimensional analysis
Autor: Da Prato, Giuseppe
Jahr: 2001
Contents
1 Gaussian measures in Hilbert spaces 1
1.1 One dimensional Hilbert spaces.1
1.2 Finite dimensional Hilbert spaces.2
1.2.1 Product probabilities.3
1.2.2 Definition of gaussian measures.3
1.3 Infinite dimensional Hilbert spaces .4
1.3.1 Some preliminaries.5
1.3.2 Definition of gaussian measures.7
1.4 Gaussian random variables.14
1.4.1 Law of a random variable .14
1.4.2 Change of variables of gaussian measures.14
1.4.3 Independent gaussian random variables.16
1.5 The reproducing kernel and the white noise function.17
2 The Cameron-Martin formula 21
2.1 Introduction and setting of the problem.21
2.2 Equivalence and singularity of product measures.22
2.3 Proof of the Cameron-Martin formula .27
2.4 The Feldman-Hajek theorem .28
3 Brownian motion 31
3.1 Construction of a Brownian motion.31
3.2 Variation of a Brownian motion.34
3.3 Wiener integral.37
3.4 Multidimensional Brownian motions.40
3.5 Law of a Brownian motion in L2(0,1).42
3.5.1 Brownian Bridge.45
vj Contents
4 Stochastic perturbations of a dynamical system 47
4.1 Introduction.47
4.2 The Ornstein-Uhlenbeck process .52
4.3 The transition semigroup in the deterministic case.53
4.4 The transition semigroup in the stochastic case .55
4.5 A generalization.52
5 Invariant measures for Markov semigroups • 65
5.1 Markov semigroups.65
5.2 Invariant measures.67
5.3 Ergodic means.71
5.4 The Von Neumann theorem.73
5.5 Ergodicity.74
5.6 Properties of the set of all invariant measures .77
6 Weak convergence of measures 61
6.1 Some additional properties of measures.61
6.2 Positive functional.84
6.3 The Prokhorov theorem.88
7 Existence and uniqueness of invariant measures 93
7.1 The Krylov-Bogoliubov theorem.93
7.1.1 Application to differential stochastic equations in R" . 97
7.2 Uniqueness of invariant measures.99
8 Examples of Markov semigroups 101
8.1 Introduction.101
8.2 The Heat semigroup .102
8.3 The Ornstein-Uhlenbeck semigroup.108
8.3.1 Regularizing property of the O. U. semigroup.110
8.3.2 Invariant Measures .113
8.4 Application to differential stochastic equations.114
9 L2 and Sobolev spaces with respect to a Gaussian measure 119
9.1 Notations.119
9.2 Orthonormal basis in L2(H,/i).120
9.2.1 The one-dimensional case.120
9.2.2 The infinite dimensional case . 122
Contents vii
9.3 Ito-Wiener decomposition.125
9.4 The Ornstein-Uhlenbeck semigroup.127
9.5 The Sobolev space W1,2(H,fi).129
9.5.1 The space L2(i7, //; H).132
9.6 The Dirichlet form associated to fi .133
9.6.1 Poincare and Log Sobolev inequalities.136
9.7 Compactness of the embedding W1,2(H,fi) C L2(//,/x) . 142
9.8 The Sobolev space IV2,2 (i/,//).145
10 Gibbs measures 149
10.1 Introduction and setting of the problem.149
10.2 Gibbs measures on L2(0,1).151
10.3 The Sobolev space W1,2(H, v).153
10.4 Symmetricity of the operator N.155
10.5 Some complements on differential stochastic equations . 157
10.5.1 Cylindrical Wiener process and stochastic convolution . 157
10.5.2 Differential stochastic equations.160
10.6 Self-adjointness of .162
A Linear Semigroups Theory 167
A.l Some preliminaries on spectral theory.167
A.2 Strongly continuous semigroups.169
A.3 The Hille-Yosida theorem.173
A.3.1 Cores.178
A.4 Dissipative operators.179
Bibliography 183 |
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| spelling | Da Prato, Giuseppe 1936-2023 Verfasser (DE-588)121352641 aut An introduction to infinite dimensional analysis Giuseppe Da Prato Pisa Scuola Normale Superiore 2001 VII, 184 S. txt rdacontent n rdamedia nc rdacarrier Appunti Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf Stochastische Analysis (DE-588)4132272-1 gnd rswk-swf Hilbert-Raum (DE-588)4159850-7 gnd rswk-swf Maßtheorie (DE-588)4074626-4 gnd rswk-swf Unendlichdimensionaler Raum (DE-588)4207852-0 gnd rswk-swf Funktionalanalysis (DE-588)4018916-8 s Unendlichdimensionaler Raum (DE-588)4207852-0 s Hilbert-Raum (DE-588)4159850-7 s DE-604 Stochastische Analysis (DE-588)4132272-1 s Maßtheorie (DE-588)4074626-4 s 1\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015440414&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Da Prato, Giuseppe 1936-2023 An introduction to infinite dimensional analysis Funktionalanalysis (DE-588)4018916-8 gnd Stochastische Analysis (DE-588)4132272-1 gnd Hilbert-Raum (DE-588)4159850-7 gnd Maßtheorie (DE-588)4074626-4 gnd Unendlichdimensionaler Raum (DE-588)4207852-0 gnd |
| subject_GND | (DE-588)4018916-8 (DE-588)4132272-1 (DE-588)4159850-7 (DE-588)4074626-4 (DE-588)4207852-0 |
| title | An introduction to infinite dimensional analysis |
| title_auth | An introduction to infinite dimensional analysis |
| title_exact_search | An introduction to infinite dimensional analysis |
| title_exact_search_txtP | An introduction to infinite dimensional analysis |
| title_full | An introduction to infinite dimensional analysis Giuseppe Da Prato |
| title_fullStr | An introduction to infinite dimensional analysis Giuseppe Da Prato |
| title_full_unstemmed | An introduction to infinite dimensional analysis Giuseppe Da Prato |
| title_short | An introduction to infinite dimensional analysis |
| title_sort | an introduction to infinite dimensional analysis |
| topic | Funktionalanalysis (DE-588)4018916-8 gnd Stochastische Analysis (DE-588)4132272-1 gnd Hilbert-Raum (DE-588)4159850-7 gnd Maßtheorie (DE-588)4074626-4 gnd Unendlichdimensionaler Raum (DE-588)4207852-0 gnd |
| topic_facet | Funktionalanalysis Stochastische Analysis Hilbert-Raum Maßtheorie Unendlichdimensionaler Raum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015440414&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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