Stochastic ordinary and stochastic partial differential equations: transition from microscopic to macroscopic equations
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
2008
|
| Series: | Stochastic modelling and applied probability
58 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 458 S. Ill. |
| ISBN: | 9780387743165 |
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MARC
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| 245 | 1 | 0 | |a Stochastic ordinary and stochastic partial differential equations |b transition from microscopic to macroscopic equations |c Peter Kotelenez |
| 264 | 1 | |a New York, NY |b Springer |c 2008 | |
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| adam_text | Contents
Introduction
Part I From Microscopic Dynamics to Mesoscopic Kinematics
1
Heuristics: Microscopic Model and Space—Time Scales
............. 9
2
Deterministic Dynamics in a Lattice Model and a Mesoscopic
(Stochastic) Limit
.............................................. 15
3
Proof of the Mesoscopic Limit Theorem
.......................... 31
Part II Mesoscopic A: Stochastic Ordinary Differential Equations
4
Stochastic Ordinary Differential Equations: Existence, Uniqueness,
and Flows Properties
........................................... 59
4.1
Preliminaries
.............................................. 59
4.2
The Governing Stochastic Ordinary Differential Equations
........ 64
4.3
Equivalence in Distribution and Flow Properties for SODEs
....... 73
4.4
Examples
................................................. 78
5
Qualitative Behavior of Correlated Brownian Motions
............. 85
5.1
Uncorrelated and Correlated Brownian Motions
................. 85
5.2
Shift and Rotational
Invariance
of w(dq,
di)
.................... 92
5.3
Separation and Magnitude of the Separation of Two Correlated
Brownian Motions with Shift-Invariant
and Frame-Indifferent Integral Kernels
......................... 94
5.4
Asymptotics of Two Correlated Brownian Motions
with Shift-Invariant and Frame-Indifferent Integral Kernels
....... 105
viii Contents
5.5
Decomposition of a Diffusion into the Flux and a Symmetric
Diffusion
.................................................110
5.6
Local Behavior of Two Correlated Brownian Motions
with Shift-Invariant and Frame-Indifferent Integral Kernels
.......116
5.7
Examples and Additional Remarks
............................121
5.8
Asymptotics of Two Correlated Brownian Motions
with Shift-Invariant Integral Kernels
...........................128
6
Proof of the Flow Property
......................................
J
33
6.
1 Proof of Statement
3
of Theorem
4.5..........................133
6.2
Smoothness of the Flow
.....................................
J
38
7
Comments on SODEs: A Comparison with Other Approaches
......151
7.1
Preliminaries and a Comparison with Kunita s Model
............151
7.2
Examples of Correlation Functions
............................156
Part III Mesoscopic B: Stochastic Partial Differential Equations
8
Stochastic Partial Differential Equations:
Finite Mass and Extensions
.....................................163
8.1
Preliminaries
..............................................163
8.2
A Priori Estimates
..........................................171
8.3
Noncoercive SPDEs
........................................174
8.4
Coercive and Noncoercive SPDEs
.............................189
8.5
General SPDEs
............................................197
8.6 Semilinear
Stochastic Partial Differential Equations
in Stratonovich Form
.......................................198
8.7
Examples
.................................................200
9
Stochastic Partial Differential Equations: Infinite Mass
............203
9.1
Noncoercive
Quasilinear
SPDEs for Infinite Mass Evolution
......203
9.2
Noncoercive
Semilinear
SPDEs for Infinite Mass Evolution
in Stratonovich Form
.......................................219
10
Stochastic Partial Differential Equations: Homogeneous
and
Isotropie
Solutions
.........................................221
11
Proof of Smoothness, Integrability,
and
Itô s
Formula
..............................................229
11.1
Basic Estimates and State Spaces
.............................229
11.2
Proof of Smoothness of
(8.25)
and
(8.73).......................246
11.3
Proof of the
Ito
formula
(8.42)................................269
12
Proof of Uniqueness
............................................273
Contents ix
13
Comments on Other Approaches to SPDEs
.......................291
13.1
Classification
..............................................291
13.1.1
Linear SPDEs
.......................................294
13.1.2
Bilinear SPDEs
......................................297
13.1.3 Semilinear
SPDEs
...................................299
13.1.4
Quasi linear SPDEs
...................................301
13.1.5
Nonlinear SPDEs
....................................301
13.1.6
Stochastic Wave Equations
............................302
13.2
Models
...................................................302
13.2.1
Nonlinear Filtering
...................................302
13.2.2
SPDEs for Mass Distributions
.........................303
13.2.3
Fluctuation Limits for Particles
.........................304
13.2.4
SPDEs in Genetics
...................................305
13.2.5
SPDEs in
Neuroscience
...............................305
13.2.6
SPDEs in Euclidean Field Theory
......................306
13.2.7
SPDEs in Fluid Mechanics
............................306
13.2.8
SPDEs in Surface Physics/Chemistry
...................308
13.2.9
SPDEs for Strings
....................................308
13.3
Books on SPDEs
...........................................308
Part IV Macroscopic: Deterministic Partial Differential Equations
14
Partial Differential Equations as a Macroscopic Limit
..............313
14.1
Limiting Equations and Hypotheses
...........................313
14.2
The Macroscopic Limit for
d
> 2.............................316
14.3
Examples
.................................................327
14.4
A Remark on
d
= 1 ........................................330
14.5
Convergence of Stochastic Transport Equations
to Macroscopic Parabolic Equations
...........................331
Part V General Appendix
15
Appendix
.....................................................335
15.1
Analysis
..................................................335
15.1.1
Metric Spaces: Extension by Continuity, Contraction
Mappings, and Uniform Boundedness
...................335
15.1.2
Some Classical Inequalities
............................336
15.1.3
The
Schwarz
Space
..................................340
15.1.4
Metrics on Spaces of Measures
.........................348
15.1.5
Riemann
Stieltjes
Integrals
............................357
15.1.6
The Skorokhod Space £>([0,
ос); В)
....................359
15.2
Stochastics
................................................362
15.2.1
Relative Compactness and Weak Convergence
............362
л
Contents
15.2.2
Regular and Cylindrical Hubert Space-Valued Brownian
Motions
............................................366
15.2.3
Martingales, Quadratic Variation, and Inequalities
.........371
15.2.4
Random Covariance and Space-time Correlations
for Correlated Brownian Motions
.......................380
15.2.5
Stochastic
Ito
Integrals
................................387
15.2.6
Stochastic Stratonovich Integrals
.......................403
15.2.7
Markov-Diffusion Processes
...........................411
15.2.8
Measure-Valued Flows: Proof of Proposition
4.3..........418
15.3
The Fractional Step Method
..................................422
15.4
Mechanics: Frame-Indifference
...............................424
Subject Index
......................................................431
Symbols
...........................................................439
References
.........................................................445
|
| adam_txt |
Contents
Introduction
Part I From Microscopic Dynamics to Mesoscopic Kinematics
1
Heuristics: Microscopic Model and Space—Time Scales
. 9
2
Deterministic Dynamics in a Lattice Model and a Mesoscopic
(Stochastic) Limit
. 15
3
Proof of the Mesoscopic Limit Theorem
. 31
Part II Mesoscopic A: Stochastic Ordinary Differential Equations
4
Stochastic Ordinary Differential Equations: Existence, Uniqueness,
and Flows Properties
. 59
4.1
Preliminaries
. 59
4.2
The Governing Stochastic Ordinary Differential Equations
. 64
4.3
Equivalence in Distribution and Flow Properties for SODEs
. 73
4.4
Examples
. 78
5
Qualitative Behavior of Correlated Brownian Motions
. 85
5.1
Uncorrelated and Correlated Brownian Motions
. 85
5.2
Shift and Rotational
Invariance
of w(dq,
di)
. 92
5.3
Separation and Magnitude of the Separation of Two Correlated
Brownian Motions with Shift-Invariant
and Frame-Indifferent Integral Kernels
. 94
5.4
Asymptotics of Two Correlated Brownian Motions
with Shift-Invariant and Frame-Indifferent Integral Kernels
. 105
viii Contents
5.5
Decomposition of a Diffusion into the Flux and a Symmetric
Diffusion
.110
5.6
Local Behavior of Two Correlated Brownian Motions
with Shift-Invariant and Frame-Indifferent Integral Kernels
.116
5.7
"Examples and Additional Remarks
.121
5.8
Asymptotics of Two Correlated Brownian Motions
with Shift-Invariant Integral Kernels
.128
6
Proof of the Flow Property
.
J
33
6.
1 Proof of Statement
3
of Theorem
4.5.133
6.2
Smoothness of the Flow
.
J
38
7
Comments on SODEs: A Comparison with Other Approaches
.151
7.1
Preliminaries and a Comparison with Kunita's Model
.151
7.2
Examples of Correlation Functions
.156
Part III Mesoscopic B: Stochastic Partial Differential Equations
8
Stochastic Partial Differential Equations:
Finite Mass and Extensions
.163
8.1
Preliminaries
.163
8.2
A Priori Estimates
.171
8.3
Noncoercive SPDEs
.174
8.4
Coercive and Noncoercive SPDEs
.189
8.5
General SPDEs
.197
8.6 Semilinear
Stochastic Partial Differential Equations
'in Stratonovich Form
.198
8.7
Examples
.200
9
Stochastic Partial Differential Equations: Infinite Mass
.203
9.1
Noncoercive
Quasilinear
SPDEs for Infinite Mass Evolution
.203
9.2
Noncoercive
Semilinear
SPDEs for Infinite Mass Evolution
in Stratonovich Form
.219
10
Stochastic Partial Differential Equations: Homogeneous
and
Isotropie
Solutions
.221
11
Proof of Smoothness, Integrability,
and
Itô's
Formula
.229
11.1
Basic Estimates and State Spaces
.229
11.2
Proof of Smoothness of
(8.25)
and
(8.73).246
11.3
Proof of the
Ito
formula
(8.42).269
12
Proof of Uniqueness
.273
Contents ix
13
Comments on Other Approaches to SPDEs
.291
13.1
Classification
.291
13.1.1
Linear SPDEs
.294
13.1.2
Bilinear SPDEs
.297
13.1.3 Semilinear
SPDEs
.299
13.1.4
Quasi linear SPDEs
.301
13.1.5
Nonlinear SPDEs
.301
13.1.6
Stochastic Wave Equations
.302
13.2
Models
.302
13.2.1
Nonlinear Filtering
.302
13.2.2
SPDEs for Mass Distributions
.303
13.2.3
Fluctuation Limits for Particles
.304
13.2.4
SPDEs in Genetics
.305
13.2.5
SPDEs in
Neuroscience
.305
13.2.6
SPDEs in Euclidean Field Theory
.306
13.2.7
SPDEs in Fluid Mechanics
.306
13.2.8
SPDEs in Surface Physics/Chemistry
.308
13.2.9
SPDEs for Strings
.308
13.3
Books on SPDEs
.308
Part IV Macroscopic: Deterministic Partial Differential Equations
14
Partial Differential Equations as a Macroscopic Limit
.313
14.1
Limiting Equations and Hypotheses
.313
14.2
The Macroscopic Limit for
d
> 2.316
14.3
Examples
.327
14.4
A Remark on
d
= 1 .330
14.5
Convergence of Stochastic Transport Equations
to Macroscopic Parabolic Equations
.331
Part V General Appendix
15
Appendix
.335
15.1
Analysis
.335
15.1.1
Metric Spaces: Extension by Continuity, Contraction
Mappings, and Uniform Boundedness
.335
15.1.2
Some Classical Inequalities
.336
15.1.3
The
Schwarz
Space
.340
15.1.4
Metrics on Spaces of Measures
.348
15.1.5
Riemann
Stieltjes
Integrals
.357
15.1.6
The Skorokhod Space £>([0,
ос); В)
.359
15.2
Stochastics
.362
15.2.1
Relative Compactness and Weak Convergence
.362
л
Contents
15.2.2
Regular and Cylindrical Hubert Space-Valued Brownian
Motions
.366
15.2.3
Martingales, Quadratic Variation, and Inequalities
.371
15.2.4
Random Covariance and Space-time Correlations
for Correlated Brownian Motions
.380
15.2.5
Stochastic
Ito
Integrals
.387
15.2.6
Stochastic Stratonovich Integrals
.403
15.2.7
Markov-Diffusion Processes
.411
15.2.8
Measure-Valued Flows: Proof of Proposition
4.3.418
15.3
The Fractional Step Method
.422
15.4
Mechanics: Frame-Indifference
.424
Subject Index
.431
Symbols
.439
References
.445 |
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| id | DE-604.BV023037889 |
| illustrated | Illustrated |
| index_date | 2024-07-02T19:19:41Z |
| indexdate | 2024-07-09T21:09:33Z |
| institution | BVB |
| isbn | 9780387743165 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016241631 |
| oclc_num | 255688142 |
| open_access_boolean | |
| owner | DE-384 DE-91G DE-BY-TUM DE-706 DE-739 DE-824 DE-11 DE-20 DE-188 |
| owner_facet | DE-384 DE-91G DE-BY-TUM DE-706 DE-739 DE-824 DE-11 DE-20 DE-188 |
| physical | X, 458 S. Ill. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Springer |
| record_format | marc |
| series | Stochastic modelling and applied probability |
| series2 | Stochastic modelling and applied probability |
| spelling | Kotelenez, Peter Verfasser aut Stochastic ordinary and stochastic partial differential equations transition from microscopic to macroscopic equations Peter Kotelenez New York, NY Springer 2008 X, 458 S. Ill. txt rdacontent n rdamedia nc rdacarrier Stochastic modelling and applied probability 58 Stochastic differential equations Stochastic partial differential equations Stochastische Differentialgeometrie (DE-588)4226826-6 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Stochastische Differentialgeometrie (DE-588)4226826-6 s DE-604 Mathematische Physik (DE-588)4037952-8 s 1\p DE-604 Stochastische Differentialgleichung (DE-588)4057621-8 s 2\p DE-604 Erscheint auch als Online-Ausgabe 978-0-387-74317-2 Stochastic modelling and applied probability 58 (DE-604)BV019623501 58 Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016241631&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Kotelenez, Peter Stochastic ordinary and stochastic partial differential equations transition from microscopic to macroscopic equations Stochastic modelling and applied probability Stochastic differential equations Stochastic partial differential equations Stochastische Differentialgeometrie (DE-588)4226826-6 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| subject_GND | (DE-588)4226826-6 (DE-588)4057621-8 (DE-588)4037952-8 |
| title | Stochastic ordinary and stochastic partial differential equations transition from microscopic to macroscopic equations |
| title_auth | Stochastic ordinary and stochastic partial differential equations transition from microscopic to macroscopic equations |
| title_exact_search | Stochastic ordinary and stochastic partial differential equations transition from microscopic to macroscopic equations |
| title_exact_search_txtP | Stochastic ordinary and stochastic partial differential equations transition from microscopic to macroscopic equations |
| title_full | Stochastic ordinary and stochastic partial differential equations transition from microscopic to macroscopic equations Peter Kotelenez |
| title_fullStr | Stochastic ordinary and stochastic partial differential equations transition from microscopic to macroscopic equations Peter Kotelenez |
| title_full_unstemmed | Stochastic ordinary and stochastic partial differential equations transition from microscopic to macroscopic equations Peter Kotelenez |
| title_short | Stochastic ordinary and stochastic partial differential equations |
| title_sort | stochastic ordinary and stochastic partial differential equations transition from microscopic to macroscopic equations |
| title_sub | transition from microscopic to macroscopic equations |
| topic | Stochastic differential equations Stochastic partial differential equations Stochastische Differentialgeometrie (DE-588)4226826-6 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| topic_facet | Stochastic differential equations Stochastic partial differential equations Stochastische Differentialgeometrie Stochastische Differentialgleichung Mathematische Physik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016241631&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV019623501 |
| work_keys_str_mv | AT kotelenezpeter stochasticordinaryandstochasticpartialdifferentialequationstransitionfrommicroscopictomacroscopicequations |