Second order PDE's in finite and infinite dimension: a probabilistic approach
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2001
|
| Series: | Lecture notes in mathematics
1762 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 330 S. |
| ISBN: | 354042136X |
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MARC
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| 100 | 1 | |a Cerrai, Sandra |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Second order PDE's in finite and infinite dimension |b a probabilistic approach |c Sandra Cerrai |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2001 | |
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Record in the Search Index
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| adam_text | Contents
Introduction 1
I Finite dimension 19
1 Kolmogorov equations in W* with unbounded coefficients 21
1.1 Assumptions 23
1.2 The solution of the associated SDE 25
1.3 Estimates for the derivatives of the solution 34
1.3.1 First derivative 37
1.3.2 Higher order derivatives 39
1.3.3 Conclusion 44
1.4 The transition semigroup 46
1.5 The derivatives of the semigroup 48
1.6 Existence and uniqueness of solutions 52
1.6.1 The parabolic case 52
1.6.2 The elliptic case 53
1.7 Schauder estimates 56
2 Asymptotic behaviour of solutions 65
2.1 Notations and preliminary results 66
2.2 Existence 70
2.3 Uniqueness 71
2.3.1 Strong Feller property 72
2.3.2 Irreducibility 74
2.4 Absolute continuity 77
2.5 The strongly dissipative case 77
3 Analyticity of the semigroup in a degenerate case 81
3.1 Assumptions 83
3.2 Some properties of the solution of the SDE 85
3.3 A generalization of the Bismut Elworthy formula 91
viii
3.4 The transition semigroup 94
3.5 The generation result 99
II Infinite dimension 103
4 Smooth dependence on data for the SPDE: the Lipschitz case 105
4.1 Notations and assumptions 107
4.1.1 The operator A 107
4.1.2 The operator Q and the stochastic convolution wA(t) .... 110
4.1.3 The Nemytskii operator 112
4.2 Differential dependence on initial data 113
4.2.1 First derivative 114
4.2.2 Higher order derivatives . 118
4.3 The transition semigroup 130
4.4 Differentiability of the transition semigroup 132
5 Kolmogorov equations in Hilbert spaces 143
5.1 Assumptions 145
5.2 The trace class property of D2(Pt p)QQ* 148
5.3 The parabolic problem 154
5.4 The elliptic problem 164
5.5 Schauder estimates 168
6 Smooth dependence on data for the SPDE: the non Lipschitz case
(I) 171
6.1 Assumptions and preliminary results 173
6.1.1 The Nemytskii operator 179
6.1.2 The approximating Nemytskii operators 181
6.1.3 Functional spaces 184
6.2 Some a priori estimates for the solution 185
6.2.1 The approximating problem 190
6.3 Differential dependence on initial data 191
6.4 Further properties of the derivatives of the solution 196
6.5 Smoothing properties of the transition semigroup 200
7 Smooth dependence on data for the SPDE: the non Lipschitz case
(II) 205
7.1 The transition semigroup 206
7.2 Some approximation results 210
7.3 Smoothing property of the transition semigroup 216
be
8 Ergodicity 221
8.1 Assumptions 222
8.2 Existence 223
8.3 Uniqueness 228
8.3.1 Strong Feller Property 229
8.3.2 Irreducibility 230
9 Hamilton Jacobi Bellman equations in Hilbert spaces 237
9.1 The state equation 239
9.2 The first variation equation 246
9.3 The approximating transition semigroups 250
9.4 The parabolic Hamilton Jacobi Bellman equation 253
9.4.1 An a priori estimate 258
9.4.2 Proof of the Theorem 9.4.2 265
9.5 The elliptic Hamilton Jacobi Bellman equation 267
9.5.1 Lipschitz hamiltonian K 269
9.5.2 Locally Lipschitz hamiltonian K 276
10 Application to stochastic optimal control problems 281
10.1 The finite horizon case 284
10.2 The infinite horizon case 289
10.3 Existence of the optimal control in the one dimensional case 297
Appendices 301
A Dissipative mappings 301
A.I Subdifferential of the norm 301
A.2 Dissipative mappings 303
B Weakly continuous semigroups 305
B.I Definition and main properties 305
B.2 Differentiability of weakly continuous semigroups 309
C Theorem of contractions depending on parameters 313
Bibliography 319
|
| any_adam_object | 1 |
| author | Cerrai, Sandra |
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| dewey-search | 519.2 |
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| dewey-tens | 510 - Mathematics |
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| indexdate | 2024-07-09T18:50:44Z |
| institution | BVB |
| isbn | 354042136X |
| language | English |
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| physical | IX, 330 S. |
| publishDate | 2001 |
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| series2 | Lecture notes in mathematics |
| spelling | Cerrai, Sandra Verfasser aut Second order PDE's in finite and infinite dimension a probabilistic approach Sandra Cerrai Berlin [u.a.] Springer 2001 IX, 330 S. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1762 Partielle Differentialgleichung - Ordnung 2 Stochastische Differentialgleichung Stochastische optimale Kontrolle Stochastic partial differential equations Stochastische optimale Kontrolle (DE-588)4207850-7 gnd rswk-swf Ordnung 2 (DE-588)4350619-7 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s Ordnung 2 (DE-588)4350619-7 s DE-604 Stochastische Differentialgleichung (DE-588)4057621-8 s Stochastische optimale Kontrolle (DE-588)4207850-7 s Lecture notes in mathematics 1762 (DE-604)BV000676446 1762 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009373905&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Cerrai, Sandra Second order PDE's in finite and infinite dimension a probabilistic approach Lecture notes in mathematics Partielle Differentialgleichung - Ordnung 2 Stochastische Differentialgleichung Stochastische optimale Kontrolle Stochastic partial differential equations Stochastische optimale Kontrolle (DE-588)4207850-7 gnd Ordnung 2 (DE-588)4350619-7 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4207850-7 (DE-588)4350619-7 (DE-588)4057621-8 (DE-588)4044779-0 |
| title | Second order PDE's in finite and infinite dimension a probabilistic approach |
| title_auth | Second order PDE's in finite and infinite dimension a probabilistic approach |
| title_exact_search | Second order PDE's in finite and infinite dimension a probabilistic approach |
| title_full | Second order PDE's in finite and infinite dimension a probabilistic approach Sandra Cerrai |
| title_fullStr | Second order PDE's in finite and infinite dimension a probabilistic approach Sandra Cerrai |
| title_full_unstemmed | Second order PDE's in finite and infinite dimension a probabilistic approach Sandra Cerrai |
| title_short | Second order PDE's in finite and infinite dimension |
| title_sort | second order pde s in finite and infinite dimension a probabilistic approach |
| title_sub | a probabilistic approach |
| topic | Partielle Differentialgleichung - Ordnung 2 Stochastische Differentialgleichung Stochastische optimale Kontrolle Stochastic partial differential equations Stochastische optimale Kontrolle (DE-588)4207850-7 gnd Ordnung 2 (DE-588)4350619-7 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Partielle Differentialgleichung - Ordnung 2 Stochastische Differentialgleichung Stochastische optimale Kontrolle Stochastic partial differential equations Ordnung 2 Partielle Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009373905&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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