Large deviations for stochastic processes:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2006]
|
| Series: | Mathematical surveys and monographs
Volume 131 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xi, 410 Seiten |
| ISBN: | 9780821841457 0821841459 9781470418700 |
Staff View
MARC
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| 100 | 1 | |a Feng, Jin |e Verfasser |0 (DE-588)130220809 |4 aut | |
| 245 | 1 | 0 | |a Large deviations for stochastic processes |c Jin Feng ; Thomas G. Kurtz |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2006] | |
| 264 | 4 | |c © 2006 | |
| 300 | |a xi, 410 Seiten | ||
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| 490 | 1 | |a Mathematical surveys and monographs |v Volume 131 | |
| 650 | 7 | |a Grandes desvios |2 larpcal | |
| 650 | 4 | |a Grandes déviations | |
| 650 | 4 | |a Markov, Processus de | |
| 650 | 7 | |a Processos estocásticos |2 larpcal | |
| 650 | 4 | |a Processus stochastiques | |
| 650 | 4 | |a Semi-groupes d'opérateurs | |
| 650 | 4 | |a Solutions de viscosité | |
| 650 | 4 | |a Large deviations | |
| 650 | 4 | |a Semigroups of operators | |
| 650 | 4 | |a Markov processes | |
| 650 | 4 | |a Stochastic processes | |
| 650 | 4 | |a Viscosity solutions | |
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Record in the Search Index
| _version_ | 1804136240340008960 |
|---|---|
| adam_text | Contents
Preface
jx
Notation xi
Introduction
1
Chapter
1.
Introduction
3
1.1.
Basic methodology
4
1.2.
The basic setting for Markov processes
6
1.3.
Related approaches
8
1.4.
Examples
10
1.5.
An outline of the major obstacles
25
Chapter
2.
An overview
29
2.1.
Basic setup
30
2.2.
Compact state spaces
30
2.3.
General state spaces
33
Part
1.
The general theory of large deviations
39
Chapter
3.
Large deviations and exponential tightness
41
3.1.
Basic definitions and results
41
3.2.
Identifying a rate function
50
3.3.
Rate functions in product spaces
53
Chapter
4.
Large deviations for stochastic processes
57
4.1.
Exponential tightness for processes
57
4.2.
Large deviations under changes of time-scale
62
4.3.
Compactification
64
4.4.
Large deviations in the compact uniform topology
65
4.5.
Exponential tightness for solutions of martingale problems
67
4.6.
Verifying compact containment
71
4.7.
Finite dimensional determination of the process rate function
73
Part
2.
Large deviations for Markov processes and semigroup
convergence
77
Chapter
5.
Large deviations for Markov processes and nonlinear semigroup
convergence
79
5.1.
Convergence of sequences of operator semigroups
79
5.2.
Applications to large deviations
82
vi
CONTENTS
Chapter
6.
Large deviations and nonlinear semigroup convergence using
viscosity solutions
97
6.1.
Viscosity solutions, definition and convergence
98
6.2.
Large deviations using viscosity semigroup convergence
106
Chapter
7.
Extensions of viscosity solution methods
109
7.1.
Viscosity solutions, definition and convergence
109
7.2.
Large deviation applications
126
7.3.
Convergence using projected operators
130
Chapter
8.
The Nisio semigroup and a control representation of the rate
function
135
8.1.
Formulation of the control problem
135
8.2.
The Nisio semigroup
141
8.3.
Control representation of the rate function
142
8.4.
Properties of the control semigroup V
143
8.5.
Verification of semigroup representation
151
8.6.
Verifying the assumptions
155
Part
3.
Examples of large deviations and the comparison principle
163
Chapter
9.
The comparison principle
165
9.1.
General estimates
165
9.2.
General conditions in Rd
172
9.3.
Bounded smooth domains in Rd with (possibly oblique) reflection
179
9.4.
Conditions for infinite dimensional state space
184
Chapter
10.
Nearly deterministic processes in Rd
199
10.1.
Processes with independent increments
199
10.2.
Random walks
207
10.3.
Markov processes
207
10.4.
Nearly deterministic Markov chains
219
10.5.
Diffusion processes with reflecting boundaries
221
Chapter
11.
Random evolutions
229
11.1.
Discrete time, law of large numbers scaling
230
11.2.
Continuous time, law of large numbers scaling
244
11.3.
Continuous time, central limit scaling
260
11.4.
Discrete time, central limit scaling
266
11.5.
Diffusions with periodic coefficients
269
11.6.
Systems with small diffusion and averaging
271
Chapter
12.
Occupation measures
283
12.1.
Occupation measures of a Markov process
-
Discrete time
284
12.2.
Occupation measures of a Markov process
-
Continuous time
288
Chapter
13.
Stochastic equations in infinite dimensions
293
13.1.
Stochastic reaction-diffusion equations on a rescaled lattice
293
13.2.
Stochastic Cahn-Hilliard equations on rescaled lattice
305
13.3.
Weakly interacting stochastic particles
315
CONTENTS
vii
Appendix
343
Appendix A. Operators and convergence in function spaces
345
A.I. Semicontinuity
345
A.
2.
General notions of convergence
346
A.3. Dissipativity of operators
350
Appendix B. Variational constants, rate of growth and spectral theory for
the semigroup of positive linear operators
353
B.I. Relationship to the spectral theory of positive operators
354
B.2. Relationship to some variational constants
357
Appendix C. Spectral properties for discrete and continuous Laplacians
367
C.I. The case of
d
= 1 368
C.2. The case of
d
> 1 368
C.3.
E
=
L2(O)
Π {ρ
:
J pdx
= 0} 369
C.
4.
Other useful approximations
370
Appendix D. Results from mass transport theory
371
D.I.
Distributional derivatives
371
D.2. Convex functions
375
D.3. The p-
Wasserstein
metric space
376
D.4. The Monge-Kantorovich problem
378
D.5. Weighted Sobolev spaces
#¿(-Rd)
and
Я 1^)
382
D.6. Fisher information and its properties
386
D.7. Mass transport inequalities
394
D.8. Miscellaneous
401
Bibliography
403
|
| adam_txt |
Contents
Preface
jx
Notation xi
Introduction
1
Chapter
1.
Introduction
3
1.1.
Basic methodology
4
1.2.
The basic setting for Markov processes
6
1.3.
Related approaches
8
1.4.
Examples
10
1.5.
An outline of the major obstacles
25
Chapter
2.
An overview
29
2.1.
Basic setup
30
2.2.
Compact state spaces
30
2.3.
General state spaces
33
Part
1.
The general theory of large deviations
39
Chapter
3.
Large deviations and exponential tightness
41
3.1.
Basic definitions and results
41
3.2.
Identifying a rate function
50
3.3.
Rate functions in product spaces
53
Chapter
4.
Large deviations for stochastic processes
57
4.1.
Exponential tightness for processes
57
4.2.
Large deviations under changes of time-scale
62
4.3.
Compactification
64
4.4.
Large deviations in the compact uniform topology
65
4.5.
Exponential tightness for solutions of martingale problems
67
4.6.
Verifying compact containment
71
4.7.
Finite dimensional determination of the process rate function
73
Part
2.
Large deviations for Markov processes and semigroup
convergence
77
Chapter
5.
Large deviations for Markov processes and nonlinear semigroup
convergence
79
5.1.
Convergence of sequences of operator semigroups
79
5.2.
Applications to large deviations
82
vi
CONTENTS
Chapter
6.
Large deviations and nonlinear semigroup convergence using
viscosity solutions
97
6.1.
Viscosity solutions, definition and convergence
98
6.2.
Large deviations using viscosity semigroup convergence
106
Chapter
7.
Extensions of viscosity solution methods
109
7.1.
Viscosity solutions, definition and convergence
109
7.2.
Large deviation applications
126
7.3.
Convergence using projected operators
130
Chapter
8.
The Nisio semigroup and a control representation of the rate
function
135
8.1.
Formulation of the control problem
135
8.2.
The Nisio semigroup
141
8.3.
Control representation of the rate function
142
8.4.
Properties of the control semigroup V
143
8.5.
Verification of semigroup representation
151
8.6.
Verifying the assumptions
155
Part
3.
Examples of large deviations and the comparison principle
163
Chapter
9.
The comparison principle
165
9.1.
General estimates
165
9.2.
General conditions in Rd
172
9.3.
Bounded smooth domains in Rd with (possibly oblique) reflection
179
9.4.
Conditions for infinite dimensional state space
184
Chapter
10.
Nearly deterministic processes in Rd
199
10.1.
Processes with independent increments
199
10.2.
Random walks
207
10.3.
Markov processes
207
10.4.
Nearly deterministic Markov chains
219
10.5.
Diffusion processes with reflecting boundaries
221
Chapter
11.
Random evolutions
229
11.1.
Discrete time, law of large numbers scaling
230
11.2.
Continuous time, law of large numbers scaling
244
11.3.
Continuous time, central limit scaling
260
11.4.
Discrete time, central limit scaling
266
11.5.
Diffusions with periodic coefficients
269
11.6.
Systems with small diffusion and averaging
271
Chapter
12.
Occupation measures
283
12.1.
Occupation measures of a Markov process
-
Discrete time
284
12.2.
Occupation measures of a Markov process
-
Continuous time
288
Chapter
13.
Stochastic equations in infinite dimensions
293
13.1.
Stochastic reaction-diffusion equations on a rescaled lattice
293
13.2.
Stochastic Cahn-Hilliard equations on rescaled lattice
305
13.3.
Weakly interacting stochastic particles
315
CONTENTS
vii
Appendix
343
Appendix A. Operators and convergence in function spaces
345
A.I. Semicontinuity
345
A.
2.
General notions of convergence
346
A.3. Dissipativity of operators
350
Appendix B. Variational constants, rate of growth and spectral theory for
the semigroup of positive linear operators
353
B.I. Relationship to the spectral theory of positive operators
354
B.2. Relationship to some variational constants
357
Appendix C. Spectral properties for discrete and continuous Laplacians
367
C.I. The case of
d
= 1 368
C.2. The case of
d
> 1 368
C.3.
E
=
L2(O)
Π {ρ
:
J pdx
= 0} 369
C.
4.
Other useful approximations
370
Appendix D. Results from mass transport theory
371
D.I.
Distributional derivatives
371
D.2. Convex functions
375
D.3. The p-
Wasserstein
metric space
376
D.4. The Monge-Kantorovich problem
378
D.5. Weighted Sobolev spaces
#¿(-Rd)
and
Я"1^)
382
D.6. Fisher information and its properties
386
D.7. Mass transport inequalities
394
D.8. Miscellaneous
401
Bibliography
403 |
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| id | DE-604.BV022246237 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:37:46Z |
| indexdate | 2024-07-09T20:53:15Z |
| institution | BVB |
| isbn | 9780821841457 0821841459 9781470418700 |
| language | English |
| lccn | 2006045899 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015457094 |
| oclc_num | 69672044 |
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| physical | xi, 410 Seiten |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | Mathematical surveys and monographs |
| series2 | Mathematical surveys and monographs |
| spelling | Feng, Jin Verfasser (DE-588)130220809 aut Large deviations for stochastic processes Jin Feng ; Thomas G. Kurtz Providence, Rhode Island American Mathematical Society [2006] © 2006 xi, 410 Seiten txt rdacontent n rdamedia nc rdacarrier Mathematical surveys and monographs Volume 131 Grandes desvios larpcal Grandes déviations Markov, Processus de Processos estocásticos larpcal Processus stochastiques Semi-groupes d'opérateurs Solutions de viscosité Large deviations Semigroups of operators Markov processes Stochastic processes Viscosity solutions Markov-Prozess (DE-588)4134948-9 gnd rswk-swf Große Abweichung (DE-588)4330658-5 gnd rswk-swf Markov-Prozess (DE-588)4134948-9 s Große Abweichung (DE-588)4330658-5 s DE-604 Kurtz, Thomas G. 1941- Verfasser (DE-588)129841714 aut Erscheint auch als Online-Ausgabe 978-1-4704-1358-3 Mathematical surveys and monographs Volume 131 (DE-604)BV000018014 131 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015457094&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Feng, Jin Kurtz, Thomas G. 1941- Large deviations for stochastic processes Mathematical surveys and monographs Grandes desvios larpcal Grandes déviations Markov, Processus de Processos estocásticos larpcal Processus stochastiques Semi-groupes d'opérateurs Solutions de viscosité Large deviations Semigroups of operators Markov processes Stochastic processes Viscosity solutions Markov-Prozess (DE-588)4134948-9 gnd Große Abweichung (DE-588)4330658-5 gnd |
| subject_GND | (DE-588)4134948-9 (DE-588)4330658-5 |
| title | Large deviations for stochastic processes |
| title_auth | Large deviations for stochastic processes |
| title_exact_search | Large deviations for stochastic processes |
| title_exact_search_txtP | Large deviations for stochastic processes |
| title_full | Large deviations for stochastic processes Jin Feng ; Thomas G. Kurtz |
| title_fullStr | Large deviations for stochastic processes Jin Feng ; Thomas G. Kurtz |
| title_full_unstemmed | Large deviations for stochastic processes Jin Feng ; Thomas G. Kurtz |
| title_short | Large deviations for stochastic processes |
| title_sort | large deviations for stochastic processes |
| topic | Grandes desvios larpcal Grandes déviations Markov, Processus de Processos estocásticos larpcal Processus stochastiques Semi-groupes d'opérateurs Solutions de viscosité Large deviations Semigroups of operators Markov processes Stochastic processes Viscosity solutions Markov-Prozess (DE-588)4134948-9 gnd Große Abweichung (DE-588)4330658-5 gnd |
| topic_facet | Grandes desvios Grandes déviations Markov, Processus de Processos estocásticos Processus stochastiques Semi-groupes d'opérateurs Solutions de viscosité Large deviations Semigroups of operators Markov processes Stochastic processes Viscosity solutions Markov-Prozess Große Abweichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015457094&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000018014 |
| work_keys_str_mv | AT fengjin largedeviationsforstochasticprocesses AT kurtzthomasg largedeviationsforstochasticprocesses |