Analysis of stochastic partial differential equations:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, R.I.
American Mathematical Society
2014
|
| Series: | CBMS regional conference series in mathematics
119 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | VIII, 116 S. graph. Darst. |
| ISBN: | 9781470415471 |
Staff View
MARC
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| 100 | 1 | |a Khoshnevisan, Davar |d 1964- |e Verfasser |0 (DE-588)124139795 |4 aut | |
| 245 | 1 | 0 | |a Analysis of stochastic partial differential equations |c Davar Khoshnevisan |
| 264 | 1 | |a Providence, R.I. |b American Mathematical Society |c 2014 | |
| 300 | |a VIII, 116 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a CBMS regional conference series in mathematics |v 119 | |
| 500 | |a Includes bibliographical references and index | ||
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| 655 | 7 | |0 (DE-588)1071861417 |a Konferenzschrift |y 2013 |z Lansing, Mich. |2 gnd-content | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Chapter
1.
Prelude
1
Chapter
2.
Wiener integrals
9
2.1.
White noise
9
2.2.
Stochastic convolutions
11
2.3.
Brownian sheet
12
2.4.
Fractional Brownian motion
15
Chapter
3.
A linear heat equation
19
3.1.
A non-random heat equation
19
3.2.
The mild solution
22
3.3.
Structure theory
22
3.4.
Approximation by interacting Brownian particles
28
3.5.
Two or more dimensions
30
3.G. Non-linear equations
30
Chapter
4.
Walsh Dalang integrals
33
4.1.
The Brownian filtration
33
4.2.
The stochastic integral
34
4.3.
Integrable
random fields
37
Chapter
5.
A non-linear heat equation
39
5.1.
Stochastic convolutions
40
5.2.
Existence and uniqueness of a mild solution
44
5.3.
Mild implies weak
50
Chapter
6.
Intermezzo: A parabolic Anderson model
53
6.1.
Brownian local times
53
6.2.
A moment bound
56
Chapter
7.
Intermittency
63
7.1.
Some motivation
63
7.2.
Intermittency and the stochastic heat equation
66
7.3.
Renewal theory
67
7.4.
Proof of Theorem
7.8 69
Chapter
8.
Intermittency fronts
71
8.1.
The problem
71
8.2.
Some proofs
72
Chapter
9.
Intermittency islands
79
9.1.
The existence and size of tall islands
79
VII
Vlil
CONTENTS
9.2.
A tail estimate
80
9.3.
On the upper bound of Theorem
9.1 82
9.4.
On the lower bound of Theorem
9.1 83
Chapter
10.
Correlation length
87
10.1.
An estimate for the length of intermittency islands
87
10.2.
A coupling for independence
89
Appendix A. Some special integrals
95
Appendix B. A Burkholder-Davis-Gundy inequality
97
Appendix C. Regularity theory
103
C.I. Garsia s theorem
103
C.2. Kolmogorov s continuity theorem
106
Bibliography 111
The general area of stochastic PDEs is interesting to mathematicians because it
contains an enormous number of challenging open problems. There is also a great
deal of interest in this topic because it has deep applications in disciplines that
range from applied mathematics, statistical mechanics, and theoretical physics, to
theoretical
neuroscience,
theory of complex chemical reactions [including polymer
science], fluid dynamics, and mathematical finance.
The stochastic PDEs that are studied in this book are similar to the familiar PDE
for heat in a thin rod, but with the additional restriction that the external forcing
density is a two-parameter stochastic process, or what is more commonly the case,
the forcing is a random noise, also known as a generalized random field. At
several points in the lectures, there are examples that highlight the phenomenon
that stochastic PDEs are not a subset of PDEs. In fact, the introduction of noise
in some partial differential equations can bring about not a small perturbation, but
truly fundamental changes to the system that the underlying PDE is attempting to
describe.
The topics covered include a brief introduction to the stochastic heat equation,
structure theory for the linear stochastic heat equation, and an in-depth look at inter-
mi ttency properties of the solution to
semilinear
stochastic heat equations. Specific
topics include stochastic integrals
à la Norbert
Wiener, an infinite-
di
men
s
і
on
al
Itô-type
stochastic integral, an example of a parabolic Anderson model, and inter-
mi ttency fronts.
There are many possible approaches to stochastic PDEs. The selection of topics and
techniques presented here are informed by the guiding example of the stochastic
heat equation.
|
| any_adam_object | 1 |
| author | Khoshnevisan, Davar 1964- |
| author_GND | (DE-588)124139795 |
| author_facet | Khoshnevisan, Davar 1964- |
| author_role | aut |
| author_sort | Khoshnevisan, Davar 1964- |
| author_variant | d k dk |
| building | Verbundindex |
| bvnumber | BV042016780 |
| classification_rvk | SI 195 SK 820 |
| classification_tum | MAT 606f |
| ctrlnum | (OCoLC)889451734 (DE-599)BSZ410405434 |
| 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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| genre | (DE-588)1071861417 Konferenzschrift 2013 Lansing, Mich. gnd-content |
| genre_facet | Konferenzschrift 2013 Lansing, Mich. |
| id | DE-604.BV042016780 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T01:10:37Z |
| institution | BVB |
| isbn | 9781470415471 |
| language | English |
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| physical | VIII, 116 S. graph. Darst. |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | CBMS regional conference series in mathematics |
| series2 | CBMS regional conference series in mathematics |
| spelling | Khoshnevisan, Davar 1964- Verfasser (DE-588)124139795 aut Analysis of stochastic partial differential equations Davar Khoshnevisan Providence, R.I. American Mathematical Society 2014 VIII, 116 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier CBMS regional conference series in mathematics 119 Includes bibliographical references and index Stochastische partielle Differentialgleichung (DE-588)4135969-0 gnd rswk-swf (DE-588)1071861417 Konferenzschrift 2013 Lansing, Mich. gnd-content Stochastische partielle Differentialgleichung (DE-588)4135969-0 s DE-604 CBMS regional conference series in mathematics 119 (DE-604)BV000004346 119 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027458546&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027458546&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Khoshnevisan, Davar 1964- Analysis of stochastic partial differential equations CBMS regional conference series in mathematics Stochastische partielle Differentialgleichung (DE-588)4135969-0 gnd |
| subject_GND | (DE-588)4135969-0 (DE-588)1071861417 |
| title | Analysis of stochastic partial differential equations |
| title_auth | Analysis of stochastic partial differential equations |
| title_exact_search | Analysis of stochastic partial differential equations |
| title_full | Analysis of stochastic partial differential equations Davar Khoshnevisan |
| title_fullStr | Analysis of stochastic partial differential equations Davar Khoshnevisan |
| title_full_unstemmed | Analysis of stochastic partial differential equations Davar Khoshnevisan |
| title_short | Analysis of stochastic partial differential equations |
| title_sort | analysis of stochastic partial differential equations |
| topic | Stochastische partielle Differentialgleichung (DE-588)4135969-0 gnd |
| topic_facet | Stochastische partielle Differentialgleichung Konferenzschrift 2013 Lansing, Mich. |
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