Random walk and the heat equation:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, RI
American Math. Soc.
2010
|
| Schriftenreihe: | Student mathematical library
55 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | IX, 156 S. graph. Darst. |
| ISBN: | 9780821848296 |
Internformat
MARC
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| 100 | 1 | |a Lawler, Gregory F. |d 1955- |e Verfasser |0 (DE-588)123908671 |4 aut | |
| 245 | 1 | 0 | |a Random walk and the heat equation |c Gregory F. Lawler |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 2010 | |
| 300 | |a IX, 156 S. |b graph. Darst. | ||
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents
Preface
vii
Chapter
1.
Random Walk and Discrete Heat Equation
1
§1.1.
Simple random walk
1
§1.2.
Boundary value problems
14
§1.3.
Heat equation
22
§1.4.
Expected time to escape
29
§1,5,
Space of harmonic functions
34
§1.6.
Exercises
40
Chapter
2,
Brownian Motion and the Heat Equation
49
§2.1.
Brownian motion
49
§2.2.
Harmonic functions
58
§2.3.
Dirichlet problem
67
§2.4.
Heat equation
73
§2.5.
Bounded domain
76
§2.6.
More on harmonic functions
85
§2.7.
Constructing Brownian motion
88
§2.8.
Exercises
92
Chapter
3.
Martingales
101
v
VI
Contents
§3.1.
Examples
101
§3.2.
Conditional expectation
108
§3.3.
Definition of martingale
112
§3.4.
Optional sampling theorem
113
§3.5.
Martingale convergence theorem
119
§3.6.
Uniform
integr
ability
123
§3.7.
Exercises
127
Chapter
4.
Fractal Dimension
135
§4.1.
Box dimension
135
§4.2.
Cantor measure
138
§4.3.
Hausdorff measure and dimension
142
§4.4.
Exercises
152
Suggestions for Further Reading
155
The heat equation can be derived by aver¬
aging over a very large number of particles.
Traditionally, the resulting PDE is studied as a
deterministic equation, an approach that has
brought many significant results and a deep
understanding of the equation and its solutions.
By studying the heat equation by considering the
individual random particles, however, one gains
further intuition into the problem. While this is now standard for
many researchers, this approach is generally not presented at the
undergraduate level. In this book, Lawler introduces the heat equa¬
tion and the closely related notion of harmonic functions from a
probabilistic perspective.
The theme of the first two chapters of the book is the relationship
between random walks and the heat equation. The first chapter
discusses the discrete case, random walk and the heat equation on
the integer lattice; and the second chapter discusses the continuous
case, Brownian motion and the usual heat equation. Relationships
are shown between the two. For example, solving the heat
équation
in the
auserete
setting becomes a problem of diagonalization of
symmetric matrices, which becomes a problem in Fourier series in
the continuous case. Random walk and Brownian motion are
intro-:
duced and developed from first prmciples. The latter two chapters
discuss different topics: martingales and fractal dimension, with the
chapters tied together by one example, a random Cantor set.
The idea of this book is to merge probabilistic and deterministic
approaches to heat flow. It is also intended as a bridge from under¬
graduate analysis to graduate and research perspectives. The book
is suitable for advanced undergraduates, particularly those consid¬
ering graduate work in mathematics or related areas.
|
| any_adam_object | 1 |
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| ctrlnum | (OCoLC)731660543 (DE-599)BVBBV037215845 |
| dewey-full | 519.2/82 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2/82 |
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| dewey-tens | 510 - Mathematics |
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| id | DE-604.BV037215845 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:53:35Z |
| institution | BVB |
| isbn | 9780821848296 |
| language | English |
| lccn | 2010031593 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-021129820 |
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| physical | IX, 156 S. graph. Darst. |
| publishDate | 2010 |
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| publisher | American Math. Soc. |
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| series | Student mathematical library |
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| spelling | Lawler, Gregory F. 1955- Verfasser (DE-588)123908671 aut Random walk and the heat equation Gregory F. Lawler Providence, RI American Math. Soc. 2010 IX, 156 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Student mathematical library 55 Irrfahrtsproblem (DE-588)4162442-7 gnd rswk-swf Wärmeleitungsgleichung (DE-588)4188859-5 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Irrfahrtsproblem (DE-588)4162442-7 s Wahrscheinlichkeitstheorie (DE-588)4079013-7 s Stochastischer Prozess (DE-588)4057630-9 s Wärmeleitungsgleichung (DE-588)4188859-5 s DE-604 Student mathematical library 55 (DE-604)BV013184751 55 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=021129820&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=021129820&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Lawler, Gregory F. 1955- Random walk and the heat equation Student mathematical library Irrfahrtsproblem (DE-588)4162442-7 gnd Wärmeleitungsgleichung (DE-588)4188859-5 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Stochastischer Prozess (DE-588)4057630-9 gnd |
| subject_GND | (DE-588)4162442-7 (DE-588)4188859-5 (DE-588)4079013-7 (DE-588)4057630-9 |
| title | Random walk and the heat equation |
| title_auth | Random walk and the heat equation |
| title_exact_search | Random walk and the heat equation |
| title_full | Random walk and the heat equation Gregory F. Lawler |
| title_fullStr | Random walk and the heat equation Gregory F. Lawler |
| title_full_unstemmed | Random walk and the heat equation Gregory F. Lawler |
| title_short | Random walk and the heat equation |
| title_sort | random walk and the heat equation |
| topic | Irrfahrtsproblem (DE-588)4162442-7 gnd Wärmeleitungsgleichung (DE-588)4188859-5 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Stochastischer Prozess (DE-588)4057630-9 gnd |
| topic_facet | Irrfahrtsproblem Wärmeleitungsgleichung Wahrscheinlichkeitstheorie Stochastischer Prozess |
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