Kolmogorov equations for stochastic PDEs:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel [u.a.]
Birkhäuser
2004
|
| Series: | Advanced courses in mathematics - CRM Barcelona
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | VII, 182 S. |
| ISBN: | 3764372168 |
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| 100 | 1 | |a Da Prato, Giuseppe |d 1936-2023 |e Verfasser |0 (DE-588)121352641 |4 aut | |
| 245 | 1 | 0 | |a Kolmogorov equations for stochastic PDEs |c Giuseppe Da Prato |
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 2004 | |
| 300 | |a VII, 182 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 0 | |a Advanced courses in mathematics - CRM Barcelona | |
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| 650 | 7 | |a Stochastische differentiaalvergelijkingen |2 gtt | |
| 650 | 4 | |a Ergodic theory | |
| 650 | 4 | |a Navier-Stokes equations | |
| 650 | 4 | |a Reaction-diffusion equations | |
| 650 | 4 | |a Stochastic analysis | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface
1
Introduction
and Preliminaries
1.1
Introduction
...............................
1.2
Preliminaries
..............................
1.2.1
Some functional spaces
.....................
1.2.2
Exponential functions
.....................
1.2.3
Gaussian measures
.......................
1.2.4
Sobolev spaces
Ψι·2(Η, μ)
and
Ψ2 2(Η,μ) .........
1.2.5
Markov semigroups
.......................
2
Stochastic Perturbations of Linear Equations
2.1
Introduction
...............................
2.2
The stochastic convolution
.......................
2.2.1
Continuity in time
.......................
2.2.2
Continuity in space and time
.................
2.2.3
The law of the stochastic convolution
............
2.3
The Ornstein-Uhlenbeck semigroup Rt
................
2.3.1
General properties
.......................
2.3.2
The infinitesimal generator of Rt
...............
2.4
The case when Rt is strong Feller
...................
2.5
Asymptotic behaviour of solutions, invariant measures
.......
2.6
The transition semigroup in
Ερ(Η,μ)
.................
2.6.1
Symmetry of Rt
........................
2.7
Poincaré
and log-Sobolev inequalities
.................
2.7.1
Hypercontractivity of Rt
....................
2.8
Some complements
...........................
2.8.1
Further regularity results when
Ą
is strong Feller
.....
2.8.2
The case when A and
С
commute
..............
2.8.3
The Ornstein-Uhlenbeck semigroup in the space of functions
of quadratic growth
......................
Contents
Stochastic Differential Equations with Lipschitz Nonlinearities
59
3.1
Introduction and setting of the problem
............... 59
3.2
Existence, uniqueness and approximation
.............. 61
3.2.1
Derivative of the solution with respect to the initial datum
64
3.3
The transition semigroup
....................... 66
3.3.1
Strong Feller property
..................... 68
3.3.2
Irreducibility
.......................... 70
3.4
Invariant measure
v
.......................... 73
3.5
The transition semigroup in L2(H, v)
................. 79
3.6
The integration by parts formula and its consequences
....... 84
3.6.1
The Sobolev space Wl>2(H,v)
................ 85
3.6.2
Poincaré
and log-Sobolev inequalities, spectral gap
..... 88
3.7
Comparison of
и
with a Gaussian measure
.............. 91
3.7.1
First method
.......................... 91
3.7.2
Second method
......................... 93
3.7.3
The adjoint of K^
....................... 97
Reaction-Diffusion Equations
99
4.1
Introduction and setting of the problem
............... 99
4.2
Solution of the stochastic differential equation
............ 102
4.3
Feller and strong Feller properties
................... 109
4.4
Irreducibility
..............................
Ill
4.5
Existence of invariant measure
.................... 114
4.5.1
The dissipative case
...................... 114
4.5.2
The non-dissipative case
.................... 115
4.6
The transition semigroup in L2(H,u)
................. 117
4.7
The integration by parts formula and its consequences
....... 122
4.7.1
The Sobolev space Wl: H,v)
................ 122
4.7.2
Poincaré
and log-Sobolev inequalities, spectral gap
..... 123
4.8
Comparison of
v
with a Gaussian measure
.............. 125
4.9
Compactness of the embedding W1 2{H,v)
С
L2 (H,
v)
....... 127
4.10
Gradient systems
............................ 129
The Stochastic Burgers Equation
131
5.1
Introduction and preliminaries
.................... 131
5.2
Solution of the stochastic differential equation
............ 135
5.3
Estimates for the solutions
....................... 138
5.4
Estimates for the derivative of the solution w.r.t. the initial datum
141
5.5
Strong Feller property and irreducibility
............... 143
5.6
Invariant measure
v
.......................... 146
5.6.1
Estimate of some integral with respect to
v
......... 147
5.7
Kolmogorov equation
.......................... 150
Contents
6
The Stochastic 2D Navier-Stokes Equation
155
6.1
Introduction and preliminaries
.................... 155
6.1.1
The abstract setting
...................... 157
6.1.2
Basic properties of the nonlinear term
............ 158
6.1.3
Sobolev embedding and
interpolatory
estimates
....... 160
6.2
Solution of the stochastic equation
.................. 161
6.3
Estimates for the solution
....................... 164
6.4
Invariant measure
v
.......................... 166
6.4.1
Estimates of some integral
................... 167
6.5
Kolmogorov equation
.......................... 168
Bibliography
173
Index
181
Giuseppe Da
Prato
Kolmogorov
Equations
for Stochastic PDEs
This book discusses stochastic partial differential equations, in particular
reaction-diffusion equations, Burgers and Navier-Stokes equations, and the
corresponding Kolmogorov equations. For each case, the transition semigroup is
considered and irreducibility, the strong Feller property, and invariant measures
are investigated. Moreover, it is proved that the exponential functions provide a
core for the infinitesimal generator. As a consequence, it is possible to study
Sobolev spaces with respect to invariant measures and to verify a basic formula
of integration by parts (the so-called
carré du
champs identity ).
Several results were proved by the author and his collaborators and appear for
the first time in book form.
Presenting the basic elements of the theory in a simple and compact way, the
book covers a one-year course directed to graduate students in mathematics or
physics. The only prerequisites are basic probability (including finite dimensional
stochastic differential equations), basic functional analysis and some elements
of the theory of partial differential equations.
ISBN
3-7643-7216-8
9 783764 372163
www.birkhauser.ch
|
| any_adam_object | 1 |
| author | Da Prato, Giuseppe 1936-2023 |
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| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.353 |
| dewey-search | 515/.353 |
| dewey-sort | 3515 3353 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-07-09T20:06:31Z |
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| language | English |
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| series2 | Advanced courses in mathematics - CRM Barcelona |
| spelling | Da Prato, Giuseppe 1936-2023 Verfasser (DE-588)121352641 aut Kolmogorov equations for stochastic PDEs Giuseppe Da Prato Basel [u.a.] Birkhäuser 2004 VII, 182 S. txt rdacontent n rdamedia nc rdacarrier Advanced courses in mathematics - CRM Barcelona Partiële differentiaalvergelijkingen gtt Stochastische differentiaalvergelijkingen gtt Ergodic theory Navier-Stokes equations Reaction-diffusion equations Stochastic analysis Stochastische partielle Differentialgleichung (DE-588)4135969-0 gnd rswk-swf Kolmogorovsche Differentialgleichungen (DE-588)4164698-8 gnd rswk-swf Stochastische partielle Differentialgleichung (DE-588)4135969-0 s Kolmogorovsche Differentialgleichungen (DE-588)4164698-8 s DE-604 Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013130297&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013130297&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Da Prato, Giuseppe 1936-2023 Kolmogorov equations for stochastic PDEs Partiële differentiaalvergelijkingen gtt Stochastische differentiaalvergelijkingen gtt Ergodic theory Navier-Stokes equations Reaction-diffusion equations Stochastic analysis Stochastische partielle Differentialgleichung (DE-588)4135969-0 gnd Kolmogorovsche Differentialgleichungen (DE-588)4164698-8 gnd |
| subject_GND | (DE-588)4135969-0 (DE-588)4164698-8 |
| title | Kolmogorov equations for stochastic PDEs |
| title_auth | Kolmogorov equations for stochastic PDEs |
| title_exact_search | Kolmogorov equations for stochastic PDEs |
| title_full | Kolmogorov equations for stochastic PDEs Giuseppe Da Prato |
| title_fullStr | Kolmogorov equations for stochastic PDEs Giuseppe Da Prato |
| title_full_unstemmed | Kolmogorov equations for stochastic PDEs Giuseppe Da Prato |
| title_short | Kolmogorov equations for stochastic PDEs |
| title_sort | kolmogorov equations for stochastic pdes |
| topic | Partiële differentiaalvergelijkingen gtt Stochastische differentiaalvergelijkingen gtt Ergodic theory Navier-Stokes equations Reaction-diffusion equations Stochastic analysis Stochastische partielle Differentialgleichung (DE-588)4135969-0 gnd Kolmogorovsche Differentialgleichungen (DE-588)4164698-8 gnd |
| topic_facet | Partiële differentiaalvergelijkingen Stochastische differentiaalvergelijkingen Ergodic theory Navier-Stokes equations Reaction-diffusion equations Stochastic analysis Stochastische partielle Differentialgleichung Kolmogorovsche Differentialgleichungen |
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