Hyperbolic dynamics and Brownian motion: an introduction
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford
Oxford Univ. Press
2012
|
| Edition: | 1. ed. |
| Series: | Oxford mathematical monographs
Oxford science publications |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 266 S. graph. Darst. |
| ISBN: | 9780199654109 |
Staff View
MARC
| LEADER | 00000nam a2200000 c 4500 | ||
|---|---|---|---|
| 001 | BV040426391 | ||
| 003 | DE-604 | ||
| 005 | 20131106 | ||
| 007 | t | ||
| 008 | 120920s2012 d||| |||| 00||| eng d | ||
| 020 | |a 9780199654109 |9 978-0-19-965410-9 | ||
| 035 | |a (OCoLC)812137426 | ||
| 035 | |a (DE-599)HBZHT017368938 | ||
| 040 | |a DE-604 |b ger |e rakwb | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-19 |a DE-11 |a DE-824 |a DE-703 |a DE-83 |a DE-20 | ||
| 082 | 0 | |a 515.39 | |
| 084 | |a SK 810 |0 (DE-625)143257: |2 rvk | ||
| 084 | |a SK 820 |0 (DE-625)143258: |2 rvk | ||
| 084 | |a SK 950 |0 (DE-625)143273: |2 rvk | ||
| 084 | |a 60Hxx |2 msc | ||
| 100 | 1 | |a Franchi, Jacques |e Verfasser |0 (DE-588)1026237327 |4 aut | |
| 245 | 1 | 0 | |a Hyperbolic dynamics and Brownian motion |b an introduction |c Jacques Franchi ; Yves Le Jan |
| 250 | |a 1. ed. | ||
| 264 | 1 | |a Oxford |b Oxford Univ. Press |c 2012 | |
| 300 | |a XIV, 266 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Oxford mathematical monographs | |
| 490 | 0 | |a Oxford science publications | |
| 650 | 0 | 7 | |a Hyperbolische Geometrie |0 (DE-588)4161041-6 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Stochastische Analysis |0 (DE-588)4132272-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Hyperbolischer Raum |0 (DE-588)4161046-5 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Hyperbolische Geometrie |0 (DE-588)4161041-6 |D s |
| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a Stochastische Analysis |0 (DE-588)4132272-1 |D s |
| 689 | 1 | |5 DE-604 | |
| 689 | 2 | 0 | |a Hyperbolischer Raum |0 (DE-588)4161046-5 |D s |
| 689 | 2 | |5 DE-604 | |
| 700 | 1 | |a Le Jan, Yves |d 1952- |e Verfasser |0 (DE-588)121457400 |4 aut | |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025278974&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-025278974 | ||
Record in the Search Index
| _version_ | 1804149487278489600 |
|---|---|
| adam_text | Contents
Introduction
xi
Summary
xiii
List of figures
xv
1
The
Lorentz-Möbius
group PSO(1, d)
1
1.1
Lie algebras and groups: introduction
1
1.1.1
M(d) and Lie subalgebras of M{d)
1
1.1.2
Basic examples of Lie algebras
3
1.1.3
The exponential map
3
1.1.4
The group associated with a Lie
subalgebra
5
1.1.5
Basic examples of Lie groups
8
1.2
Minkowski space and its pseudo-metric
10
1.3
The
Lorentz-Möbius
group and its Lie algebra
12
1.4
Two remarkable subgroups of PSO(1,
ci)
14
1.5
Structure of the elements of PSO(1, d)
16
1.6
The hyperbolic space
ШЃ
and its boundary dMd
20
1.7
The Cartan and Iwasawa decompositions of PSO(l,d)
21
1.8
Notes and comments
22
2
Hyperbolic geometry
23
2.1
The hyperbolic metric
23
2.2
Geodesies and light rays
25
2.2.1
Hyperbolic geodesies
25
2.2.2
Projection onto a light ray, and tangent bundle
26
2.2.3
Harmonic conjugation
29
2.3
Flows and leaves
33
2.4
Physical interpretations
37
2.4.1
Change of frame and relative velocities
37
2.4.2
Motion of particles
39
2.4.3
Geodesies
39
2.4.4
Horospheres
40
2.5
Poincaré
ball and half-space models
41
2.5.1 Stereographic
projection and the ball model
41
2.5.2
Upper-half-space model and the
Poincaré
coordinates
43
2.6
A commutation relation
44
2.7
The Busemann function
48
2.8
Notes and comments
50
viii Contents
3 Operators
and measures
51
3.1
The
Casimir
operator on PSO(1, d)
51
3.2
The Laplace operator
53
3.3 Haar
measure of PSO(l,d)
55
3.4
The spherical Laplacian
Δδ
63
3.5
The hyperbolic Laplacian
Δ
66
3.6
Harmonic, Liouville and volume measures
69
3.6.1
Harmonic measures and the
Poisson
kernel
69
3.6.2
Liouville and volume measures
72
3.7
Notes and comments
78
4
Kleinian groups
80
4.1
Terminology
80
4.2
Dirichlet polyhedra
81
4.3
Parabolic tessellation by an ideal 2n-gon
82
4.4
Examples of modular groups
87
4.4.1
Plane tessellation by means of two parabolic isometries
87
4.4.2
From
Г(2)
to
Г(1)
94
4.4.3
Plane tessellation by means of two boosts, and DT(l)
97
4.4.4
Plane tessellation yielding
Г(3)
101
4.5
Notes and comments
103
5
Measures and flows on T ¥d
105
5.1
Measures of
Г
-invariant
sets
105
5.2
Ergodicity
107
5.3
A mixing theorem
109
5.4
Poincaré
inequalities
112
5.4.1
The Euclidean case
113
5.4.2
The case of a fundamental domain in Hd
118
5.5
Notes and comments
123
6
Basic
Ito
calculus
124
6.1
Discrete martingales and stochastic integrals
124
6.2
Brownian motion
127
6.3
Martingales in continuous time
128
6.4
The
Ito
integral
132
6.5
Itô s
formula
135
6.6
The Stratonovich integral
144
6.7
Notes and comments
145
7
Brownian motions on groups of matrices
146
7.1
Stochastic differential equations
146
7.2
Linear stochastic differential equations
150
7.3
Approximation of a left Brownian motion by exponentials
161
7.4
Lyapunov exponents
168
Contents ix
7.5 Diffusion
processes
170
7.6
Examples of group-valued
Brownian
motions
172
7.6.1
Exponential
semimartingale
172
7.6.2
Left Brownian motion on the
Heisenberg
group
Ή.3
172
7.6.3
Brownian motions in SL(2)
175
7.6.4
Left Brownian motion on SO(d)
176
7.6.5
Left Brownian motions on PSO(1, d) and Ad
178
7.6.6
Spherical Brownian motion
182
7.6.7
Hyperbolic Brownian motion
184
7.7
Relativistic diffusion
187
7.7.1
Left Brownian motion on the
Poincaré
group T>d+X
187
7.7.2
Asymptotic behaviour of the relativistic diffusion
188
7.8
Notes and comments
192
8
The central limit theorem for geodesies
194
8.1
Dual Ad-valued left Brownian motions
195
8.2
Two dual diffusions
199
8.3
Spectral gap along the foliation
201
8.4
The resolvent kernel and conjugate functions
206
8.4.1
Differential 1-forms on Fd
206
8.4.2
Lift of
ƒ
to the 1-form wf
207
8.5
Contour deformation
208
8.6
The divergence of
ω?
211
8.7
Sinai s central limit theorem
215
8.8
Notes and comments
223
Appendix A: Geometry
224
A.I Structure of pseudo-symmetric matrices
224
A.2 The full commutation relation in PSO(l,d)
228
A.3 The d Alembertian
D
on Rhd
233
A.4 Core-cusp decomposition
236
Appendix B: Stochastic calculus
239
B.I A simple construction of a real Brownian motion
239
B.2 Chaos expansion
242
B.3 Brownian path and limiting geodesic
245
References
249
General notation
255
Other notation
256
Index of terms
261
|
| any_adam_object | 1 |
| author | Franchi, Jacques Le Jan, Yves 1952- |
| author_GND | (DE-588)1026237327 (DE-588)121457400 |
| author_facet | Franchi, Jacques Le Jan, Yves 1952- |
| author_role | aut aut |
| author_sort | Franchi, Jacques |
| author_variant | j f jf j y l jy jyl |
| building | Verbundindex |
| bvnumber | BV040426391 |
| classification_rvk | SK 810 SK 820 SK 950 |
| ctrlnum | (OCoLC)812137426 (DE-599)HBZHT017368938 |
| dewey-full | 515.39 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.39 |
| dewey-search | 515.39 |
| dewey-sort | 3515.39 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. ed. |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01900nam a2200481 c 4500</leader><controlfield tag="001">BV040426391</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20131106 </controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">120920s2012 d||| |||| 00||| eng d</controlfield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9780199654109</subfield><subfield code="9">978-0-19-965410-9</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)812137426</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)HBZHT017368938</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-19</subfield><subfield code="a">DE-11</subfield><subfield code="a">DE-824</subfield><subfield code="a">DE-703</subfield><subfield code="a">DE-83</subfield><subfield code="a">DE-20</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">515.39</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 810</subfield><subfield code="0">(DE-625)143257:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 820</subfield><subfield code="0">(DE-625)143258:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 950</subfield><subfield code="0">(DE-625)143273:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">60Hxx</subfield><subfield code="2">msc</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Franchi, Jacques</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)1026237327</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Hyperbolic dynamics and Brownian motion</subfield><subfield code="b">an introduction</subfield><subfield code="c">Jacques Franchi ; Yves Le Jan</subfield></datafield><datafield tag="250" ind1=" " ind2=" "><subfield code="a">1. ed.</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Oxford</subfield><subfield code="b">Oxford Univ. Press</subfield><subfield code="c">2012</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">XIV, 266 S.</subfield><subfield code="b">graph. Darst.</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="0" ind2=" "><subfield code="a">Oxford mathematical monographs</subfield></datafield><datafield tag="490" ind1="0" ind2=" "><subfield code="a">Oxford science publications</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Hyperbolische Geometrie</subfield><subfield code="0">(DE-588)4161041-6</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Stochastische Analysis</subfield><subfield code="0">(DE-588)4132272-1</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Hyperbolischer Raum</subfield><subfield code="0">(DE-588)4161046-5</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Hyperbolische Geometrie</subfield><subfield code="0">(DE-588)4161041-6</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="689" ind1="1" ind2="0"><subfield code="a">Stochastische Analysis</subfield><subfield code="0">(DE-588)4132272-1</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="1" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="689" ind1="2" ind2="0"><subfield code="a">Hyperbolischer Raum</subfield><subfield code="0">(DE-588)4161046-5</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="2" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Le Jan, Yves</subfield><subfield code="d">1952-</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)121457400</subfield><subfield code="4">aut</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">Digitalisierung UB Bayreuth</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025278974&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-025278974</subfield></datafield></record></collection> |
| id | DE-604.BV040426391 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:23:48Z |
| institution | BVB |
| isbn | 9780199654109 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025278974 |
| oclc_num | 812137426 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-11 DE-824 DE-703 DE-83 DE-20 |
| owner_facet | DE-19 DE-BY-UBM DE-11 DE-824 DE-703 DE-83 DE-20 |
| physical | XIV, 266 S. graph. Darst. |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | Oxford Univ. Press |
| record_format | marc |
| series2 | Oxford mathematical monographs Oxford science publications |
| spelling | Franchi, Jacques Verfasser (DE-588)1026237327 aut Hyperbolic dynamics and Brownian motion an introduction Jacques Franchi ; Yves Le Jan 1. ed. Oxford Oxford Univ. Press 2012 XIV, 266 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Oxford mathematical monographs Oxford science publications Hyperbolische Geometrie (DE-588)4161041-6 gnd rswk-swf Stochastische Analysis (DE-588)4132272-1 gnd rswk-swf Hyperbolischer Raum (DE-588)4161046-5 gnd rswk-swf Hyperbolische Geometrie (DE-588)4161041-6 s DE-604 Stochastische Analysis (DE-588)4132272-1 s Hyperbolischer Raum (DE-588)4161046-5 s Le Jan, Yves 1952- Verfasser (DE-588)121457400 aut Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025278974&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Franchi, Jacques Le Jan, Yves 1952- Hyperbolic dynamics and Brownian motion an introduction Hyperbolische Geometrie (DE-588)4161041-6 gnd Stochastische Analysis (DE-588)4132272-1 gnd Hyperbolischer Raum (DE-588)4161046-5 gnd |
| subject_GND | (DE-588)4161041-6 (DE-588)4132272-1 (DE-588)4161046-5 |
| title | Hyperbolic dynamics and Brownian motion an introduction |
| title_auth | Hyperbolic dynamics and Brownian motion an introduction |
| title_exact_search | Hyperbolic dynamics and Brownian motion an introduction |
| title_full | Hyperbolic dynamics and Brownian motion an introduction Jacques Franchi ; Yves Le Jan |
| title_fullStr | Hyperbolic dynamics and Brownian motion an introduction Jacques Franchi ; Yves Le Jan |
| title_full_unstemmed | Hyperbolic dynamics and Brownian motion an introduction Jacques Franchi ; Yves Le Jan |
| title_short | Hyperbolic dynamics and Brownian motion |
| title_sort | hyperbolic dynamics and brownian motion an introduction |
| title_sub | an introduction |
| topic | Hyperbolische Geometrie (DE-588)4161041-6 gnd Stochastische Analysis (DE-588)4132272-1 gnd Hyperbolischer Raum (DE-588)4161046-5 gnd |
| topic_facet | Hyperbolische Geometrie Stochastische Analysis Hyperbolischer Raum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025278974&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT franchijacques hyperbolicdynamicsandbrownianmotionanintroduction AT lejanyves hyperbolicdynamicsandbrownianmotionanintroduction |