Stochastic differential equations in infinite dimensions: with applications to stochastic partial differential equations
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
Berlin
Springer
[2011]
|
| Series: | Probability and its applications
|
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | xvi, 291 Seiten |
| ISBN: | 9783642161933 |
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MARC
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Record in the Search Index
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| adam_text |
IMAGE 1
CONTENTS
PART I STOCHASTIC DIFFERENTIAL EQUATIONS IN INFINITE DIMENSIONS 1
PARTIAL DIFFERENTIAL EQUATIONS AS EQUATIONS IN INFINITE DIMENSIONS . 3
1.1 THE HEAT EQUATION AS AN ABSTRACT CAUCHY PROBLEM 3
1.2 ELEMENTS OF SEMIGROUP THEORY 5
1.3 COMMONLY USED FUNCTION SPACES 10
1.4 THE ABSTRACT CAUCHY PROBLEM 11
1.5 THE VARIATIONAL METHOD 15
2 STOCHASTIC CALCULUS 17
2.1 HILBERT-SPACE-VALUED PROCESS, MARTINGALES, AND CYLINDRICAL WIENER
PROCESS 17
2.1.1 CYLINDRICAL AND HILBERT-SPACE-VALUED GAUSSIAN RANDOM VARIABLES 17
2.1.2 CYLINDRICAL AND Q-WIENER PROCESSES 19
2.1.3 MARTINGALES IN A HILBERT SPACE 21
2.2 STOCHASTIC INTEGRAL WITH RESPECT TO A WIENER PROCESS 23 2.2.1
ELEMENTARY PROCESSES 25
2.2.2 STOCHASTIC ITO INTEGRAL FOR ELEMENTARY PROCESSES 26 2.2.3
STOCHASTIC ITO INTEGRAL WITH RESPECT TO A Q-WIENER PROCESS 34 2.2.4
STOCHASTIC ITO INTEGRAL WITH RESPECT TO CYLINDRICAL WIENER PROCESS 44
2.2.5 THE MARTINGALE REPRESENTATION THEOREM 49
2.2.6 STOCHASTIC FUBINI THEOREM 57
2.3 THE ITO FORMULA 61
2.3.1 THE CASE OF A SS-WIENER PROCESS 61
2.3.2 THE CASE OF A CYLINDRICAL WIENER PROCESS 69
3 STOCHASTIC DIFFERENTIAL EQUATIONS 73
3.1 STOCHASTIC DIFFERENTIAL EQUATIONS AND THEIR SOLUTIONS 73 3.2
SOLUTIONS UNDER LIPSCHITZ CONDITIONS 84
3.3 A SPECIAL CASE 103
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1006508031
DIGITALISIERT DURCH
IMAGE 2
CONTENTS
3.4 MARKOV PROPERTY AND UNIQUENESS 107
3.5 DEPENDENCE OF THE SOLUTION ON THE INITIAL VALUE I LL
3.6 KOLMOGOROV'S BACKWARD EQUATION 117
3.7 LIPSCHITZ-TYPE APPROXIMATION OF CONTINUOUS COEFFICIENTS 125 3.8
EXISTENCE OF WEAK SOLUTIONS UNDER CONTINUITY ASSUMPTION . . . 128 3.9
COMPACT SEMIGROUPS AND EXISTENCE OF MARTINGALE SOLUTIONS . . . . 135
3.10 MILD SOLUTIONS TO SSDES DRIVEN BY CYLINDRICAL WIENER PROCESS . .
141
APPENDIX: COMPACTNESS AND TIGHTNESS OF MEASURES 148
4 SOLUTIONS BY VARIATIONAL METHOD 151
4.1 INTRODUCTION 151
4.2 EXISTENCE OF WEAK SOLUTIONS UNDER COMPACT EMBEDDING 152 4.3 STRONG
VARIATIONAL SOLUTIONS 174
4.4 MARKOV AND STRONG MARKOV PROPERTIES 181
5 STOCHASTIC DIFFERENTIAL EQUATIONS WITH DISCONTINUOUS DRIFT 185 5.1
INTRODUCTION 185
5.2 UNBOUNDED SPIN SYSTEMS, SOLUTIONS IN C([0, T], H W ) 185 5.3 LOCALLY
INTERACTING PARTICLE SYSTEMS, SOLUTIONS ],R ZRF ) 194
PART II STABILITY, BOUNDEDNESS, AND INVARIANT MEASURES
6 STABILITY THEORY FOR STRONG AND MILD SOLUTIONS 203
6.1 INTRODUCTION 203
6.2 EXPONENTIAL STABILITY FOR STOCHASTIC DIFFERENTIAL EQUATIONS 211 6.3
STABILITY IN THE VARIATIONAL METHOD 225
APPENDIX: STOCHASTIC ANALOGUE OF THE DATKO THEOREM 230
7 ULTIMATE BOUNDEDNESS AND INVARIANT MEASURE 233
7.1 EXPONENTIAL ULTIMATE BOUNDEDNESS IN THE M.S.S 233
7.2 EXPONENTIAL ULTIMATE BOUNDEDNESS IN VARIATIONAL METHOD 237 7.3
ABSTRACT CAUCHY PROBLEM, STABILITY AND EXPONENTIAL ULTIMATE BOUNDEDNESS
246
7.4 ULTIMATE BOUNDEDNESS AND INVARIANT MEASURE 255
7.4.1 VARIATIONAL EQUATIONS 261
7.4.2 SEMILINEAR EQUATIONS DRIVEN BY A SS-WIENER PROCESS . . . 266 7.4.3
SEMILINEAR EQUATIONS DRIVEN BY A CYLINDRICAL WIENER PROCESS 271
7.5 ULTIMATE BOUNDEDNESS AND WEAK RECURRENCE OF THE SOLUTIONS . . . 274
REFERENCES 285
INDEX 289 |
| any_adam_object | 1 |
| author | Gawarecki, Leszek Mandrekar, Vidyadhar 1939- |
| author_GND | (DE-588)1126283908 (DE-588)1013611721 |
| author_facet | Gawarecki, Leszek Mandrekar, Vidyadhar 1939- |
| author_role | aut aut |
| author_sort | Gawarecki, Leszek |
| author_variant | l g lg v m vm |
| building | Verbundindex |
| bvnumber | BV037219689 |
| classification_rvk | SK 820 |
| ctrlnum | (OCoLC)845953241 (DE-599)DNB1006508031 |
| dewey-full | 519.22 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.22 |
| dewey-search | 519.22 |
| dewey-sort | 3519.22 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| isbn | 9783642161933 |
| language | English |
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| physical | xvi, 291 Seiten |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Springer |
| record_format | marc |
| series2 | Probability and its applications |
| spelling | Gawarecki, Leszek Verfasser (DE-588)1126283908 aut Stochastic differential equations in infinite dimensions with applications to stochastic partial differential equations Leszek Gawarecki, Vidyadhar Mandrekar Berlin Springer [2011] © 2011 xvi, 291 Seiten txt rdacontent n rdamedia nc rdacarrier Probability and its applications Dimension unendlich (DE-588)4474010-4 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 s Dimension unendlich (DE-588)4474010-4 s DE-604 Mandrekar, Vidyadhar 1939- Verfasser (DE-588)1013611721 aut Erscheint auch als Online-Ausgabe 978-3-642-16194-0 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3532045&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021133597&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gawarecki, Leszek Mandrekar, Vidyadhar 1939- Stochastic differential equations in infinite dimensions with applications to stochastic partial differential equations Dimension unendlich (DE-588)4474010-4 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd |
| subject_GND | (DE-588)4474010-4 (DE-588)4057621-8 |
| title | Stochastic differential equations in infinite dimensions with applications to stochastic partial differential equations |
| title_auth | Stochastic differential equations in infinite dimensions with applications to stochastic partial differential equations |
| title_exact_search | Stochastic differential equations in infinite dimensions with applications to stochastic partial differential equations |
| title_full | Stochastic differential equations in infinite dimensions with applications to stochastic partial differential equations Leszek Gawarecki, Vidyadhar Mandrekar |
| title_fullStr | Stochastic differential equations in infinite dimensions with applications to stochastic partial differential equations Leszek Gawarecki, Vidyadhar Mandrekar |
| title_full_unstemmed | Stochastic differential equations in infinite dimensions with applications to stochastic partial differential equations Leszek Gawarecki, Vidyadhar Mandrekar |
| title_short | Stochastic differential equations in infinite dimensions |
| title_sort | stochastic differential equations in infinite dimensions with applications to stochastic partial differential equations |
| title_sub | with applications to stochastic partial differential equations |
| topic | Dimension unendlich (DE-588)4474010-4 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd |
| topic_facet | Dimension unendlich Stochastische Differentialgleichung |
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