Lectures on white noise functionals:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New Jersey [u.a.]
World Scientific
2008
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references |
| Physical Description: | XIII, 266 S. graph. Darst. |
| ISBN: | 9789812560520 9812560521 |
Staff View
MARC
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| 100 | 1 | |a Hida, Takeyuki |d 1927-2017 |e Verfasser |0 (DE-588)124605257 |4 aut | |
| 245 | 1 | 0 | |a Lectures on white noise functionals |c T. Hida ; Si Si |
| 264 | 1 | |a New Jersey [u.a.] |b World Scientific |c 2008 | |
| 300 | |a XIII, 266 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references | ||
| 650 | 4 | |a White noise theory | |
| 650 | 4 | |a Gaussian processes | |
| 650 | 4 | |a Gaussian processes | |
| 650 | 4 | |a White noise theory | |
| 650 | 0 | 7 | |a Weißes Rauschen |0 (DE-588)4189502-2 |2 gnd |9 rswk-swf |
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| 700 | 1 | |a Si, Si |e Verfasser |0 (DE-588)1168187079 |4 aut | |
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Record in the Search Index
| _version_ | 1804137696574046208 |
|---|---|
| adam_text | Contents
Preface vii
1.
Introduction
1
1.1
Preliminaries
........................ 1
1.2
Our idea of establishing white noise analysis
....... 2
1.3
A brief synopsis of the book
................ 6
1.4
Some general background
................. 8
1.4.1
Characteristics of white noise analysis
....... 10
2.
Generalized white noise functionals
13
2.1
Brownian motion and
Poisson
process; elemental
stochastic processes
..................... 13
2.2
Comparison between Brownian motion and
Poisson
process
............................ 21
2.3
The Bochner-Minlos theorem
............... 22
2.4
Observation of white noise through the Levy s
construction of Brownian motion
............. 26
2.5
Spaces (L2),
F
and
Τ
arising from white noise
...... 27
2.6
Generalized white noise functionals
............ 35
2.7
Creation and annihilation operators
............ 50
2.8
Examples
.......................... 54
2.9
Addenda
........................... 57
3.
Elemental random variables and Gaussian processes
63
3.1
Elemental noises
...................... 63
3.2
Canonical representation of a Gaussian process
..... 70
xii
Lectures on
White
Noise Functionals
3.3
Multiple Markov Gaussian processes
........... 81
3.4
Fractional Brownian motion
................ 86
3.5
Stationarity of fractional Brownian motion
........ 91
3.6
Fractional order differential operator in connection with
Levy s Brownian motion
.................. 95
3.7
Gaussian random fields
................... 97
4.
Linear processes and linear fields
99
4.1
Gaussian systems
...................... 100
4.2
Poisson
systems
....................... 107
4.3
Linear functionals of
Poisson
noise
............ 108
4.4
Linear processes
....................... 109
4.5
Levy field and generalized Levy field
........... 113
4.6
Gaussian elemental noises
................. 114
5.
Harmonic analysis arising from infinite dimensional
rotation group
115
5.1
Introduction
......................... 115
5.2
Infinite dimensional rotation group O(E)
......... 117
5.3
Harmonic analysis
...................... 120
5.4
Addenda to the diagram
.................. 126
5.5
The Levy group, the Windmill subgroup and the
sign-changing subgroup of O(E)
.............. 128
5.6
Classification of rotations in O(E)
............ 136
5.7
Unitary representation of the infinite dimensional
rotation group O(E)
.................... 139
5.8
Laplacian
.......................... 140
6.
Complex white noise and infinite dimensional unitary group
153
6.1
Why complex?
....................... 153
6.2
Some background
...................... 154
6.3
Subgroups of U{EC)
..................... 159
6.4
Applications
......................... 170
7.
Characterization of
Poisson
noise
175
7.1
Preliminaries
........................ 175
7.2
A characteristic of
Poisson
noise
.............. 178
7.3
A characterization of
Poisson
noise
............ 186
Contents xiii
7.4
Comparison of two noises; Gaussian and
Poisson
.... 191
7.5
Poisson
noise functionals
.................. 194
8.
Innovation theory
197
8.1
A short history of innovation theory
........... 198
8.2
Definitions and examples
.................. 200
8.3
Innovations in the weak sense
............... 204
8.4
Some other concrete examples
............... 208
9.
Variational calculus for random fields and operator fields
211
9.1
Introduction
......................... 211
9.2
Stochastic variational equations
.............. 212
9.3
Illustrative examples
.................... 213
9.4
Integrals of operators
.................... 216
9.4.1
Operators of linear form
.............. 216
9.4.2
Operators of quadratic forms of the creation and
the annihilation operators
............. 217
9.4.3
Polynomials in dt, d*; t,s€R, of degree
2..... 220
10.
Four notable roads to quantum dynamics
223
10.1
White noise approach to path integrals
.......... 223
10.2
Hamiltonian dynamics and Chern-Simons functional
integral
............................ 230
10.3
Dirichlet forms
....................... 234
10.4
Time operator
........................ 239
10.5
Addendum: Euclidean fields
................ 248
Appendix
249
Bibliography
253
Subject Index
263
|
| adam_txt |
Contents
Preface vii
1.
Introduction
1
1.1
Preliminaries
. 1
1.2
Our idea of establishing white noise analysis
. 2
1.3
A brief synopsis of the book
. 6
1.4
Some general background
. 8
1.4.1
Characteristics of white noise analysis
. 10
2.
Generalized white noise functionals
13
2.1
Brownian motion and
Poisson
process; elemental
stochastic processes
. 13
2.2
Comparison between Brownian motion and
Poisson
process
. 21
2.3
The Bochner-Minlos theorem
. 22
2.4
Observation of white noise through the Levy's
construction of Brownian motion
. 26
2.5
Spaces (L2),
F
and
Τ
arising from white noise
. 27
2.6
Generalized white noise functionals
. 35
2.7
Creation and annihilation operators
. 50
2.8
Examples
. 54
2.9
Addenda
. 57
3.
Elemental random variables and Gaussian processes
63
3.1
Elemental noises
. 63
3.2
Canonical representation of a Gaussian process
. 70
xii
Lectures on
White
Noise Functionals
3.3
Multiple Markov Gaussian processes
. 81
3.4
Fractional Brownian motion
. 86
3.5
Stationarity of fractional Brownian motion
. 91
3.6
Fractional order differential operator in connection with
Levy's Brownian motion
. 95
3.7
Gaussian random fields
. 97
4.
Linear processes and linear fields
99
4.1
Gaussian systems
. 100
4.2
Poisson
systems
. 107
4.3
Linear functionals of
Poisson
noise
. 108
4.4
Linear processes
. 109
4.5
Levy field and generalized Levy field
. 113
4.6
Gaussian elemental noises
. 114
5.
Harmonic analysis arising from infinite dimensional
rotation group
115
5.1
Introduction
. 115
5.2
Infinite dimensional rotation group O(E)
. 117
5.3
Harmonic analysis
. 120
5.4
Addenda to the diagram
. 126
5.5
The Levy group, the Windmill subgroup and the
sign-changing subgroup of O(E)
. 128
5.6
Classification of rotations in O(E)
. 136
5.7
Unitary representation of the infinite dimensional
rotation group O(E)
. 139
5.8
Laplacian
. 140
6.
Complex white noise and infinite dimensional unitary group
153
6.1
Why complex?
. 153
6.2
Some background
. 154
6.3
Subgroups of U{EC)
. 159
6.4
Applications
. 170
7.
Characterization of
Poisson
noise
175
7.1
Preliminaries
. 175
7.2
A characteristic of
Poisson
noise
. 178
7.3
A characterization of
Poisson
noise
. 186
Contents xiii
7.4
Comparison of two noises; Gaussian and
Poisson
. 191
7.5
Poisson
noise functionals
. 194
8.
Innovation theory
197
8.1
A short history of innovation theory
. 198
8.2
Definitions and examples
. 200
8.3
Innovations in the weak sense
. 204
8.4
Some other concrete examples
. 208
9.
Variational calculus for random fields and operator fields
211
9.1
Introduction
. 211
9.2
Stochastic variational equations
. 212
9.3
Illustrative examples
. 213
9.4
Integrals of operators
. 216
9.4.1
Operators of linear form
. 216
9.4.2
Operators of quadratic forms of the creation and
the annihilation operators
. 217
9.4.3
Polynomials in dt, d*; t,s€R, of degree
2. 220
10.
Four notable roads to quantum dynamics
223
10.1
White noise approach to path integrals
. 223
10.2
Hamiltonian dynamics and Chern-Simons functional
integral
. 230
10.3
Dirichlet forms
. 234
10.4
Time operator
. 239
10.5
Addendum: Euclidean fields
. 248
Appendix
249
Bibliography
253
Subject Index
263 |
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| dewey-full | 519.2/2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2/2 |
| dewey-search | 519.2/2 |
| dewey-sort | 3519.2 12 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
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| id | DE-604.BV023342847 |
| illustrated | Illustrated |
| index_date | 2024-07-02T21:02:15Z |
| indexdate | 2024-07-09T21:16:24Z |
| institution | BVB |
| isbn | 9789812560520 9812560521 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016526590 |
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| physical | XIII, 266 S. graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
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| publisher | World Scientific |
| record_format | marc |
| spelling | Hida, Takeyuki 1927-2017 Verfasser (DE-588)124605257 aut Lectures on white noise functionals T. Hida ; Si Si New Jersey [u.a.] World Scientific 2008 XIII, 266 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references White noise theory Gaussian processes Weißes Rauschen (DE-588)4189502-2 gnd rswk-swf Gauß-Prozess (DE-588)4156111-9 gnd rswk-swf Weißes Rauschen (DE-588)4189502-2 s Gauß-Prozess (DE-588)4156111-9 s DE-604 Si, Si Verfasser (DE-588)1168187079 aut Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016526590&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hida, Takeyuki 1927-2017 Si, Si Lectures on white noise functionals White noise theory Gaussian processes Weißes Rauschen (DE-588)4189502-2 gnd Gauß-Prozess (DE-588)4156111-9 gnd |
| subject_GND | (DE-588)4189502-2 (DE-588)4156111-9 |
| title | Lectures on white noise functionals |
| title_auth | Lectures on white noise functionals |
| title_exact_search | Lectures on white noise functionals |
| title_exact_search_txtP | Lectures on white noise functionals |
| title_full | Lectures on white noise functionals T. Hida ; Si Si |
| title_fullStr | Lectures on white noise functionals T. Hida ; Si Si |
| title_full_unstemmed | Lectures on white noise functionals T. Hida ; Si Si |
| title_short | Lectures on white noise functionals |
| title_sort | lectures on white noise functionals |
| topic | White noise theory Gaussian processes Weißes Rauschen (DE-588)4189502-2 gnd Gauß-Prozess (DE-588)4156111-9 gnd |
| topic_facet | White noise theory Gaussian processes Weißes Rauschen Gauß-Prozess |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016526590&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT hidatakeyuki lecturesonwhitenoisefunctionals AT sisi lecturesonwhitenoisefunctionals |