Denumerable Markov chains: generating functions, boundary theory, random walks on trees
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| Main Author: | |
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| Format: | Book |
| Language: | English Italian |
| Published: |
Zürich
Europ. Math. Soc.
2009
|
| Series: | EMS textbooks in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVII, 351 S. graph. Darst. |
| ISBN: | 9783037190715 303719071X |
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MARC
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| 245 | 1 | 0 | |a Denumerable Markov chains |b generating functions, boundary theory, random walks on trees |c Wolfgang Woess |
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Record in the Search Index
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| adam_text | Contents
Preface
v
Introduction ix
Summary .... ...... . . . . . . . .... . . . .... ... ix
Raison d’etre .................................... xiii
1 Preliminaries and basic facts 1
A Preliminaries, examples ............. . . . .... . ... 1
B Axiomatic definition of a Markov chain ........................ 5
C Transition probabilities in n steps......................... 12
D Generating functions of transition probabilities.......... 17
2 Irreducible classes 28
A Irreducible and essential classes . . . .. . . . . ........ . 28
B The period of an irreducible class......... 35
C The spectral radius of an irreducible class .............. 39
3 Recurrence and transience, convergence, and the ergodic theorem 43
A Recurrent classes ......................................... . 43
B Return times, positive recurrence, and stationary probability measures 47
C The convergence theorem for finite Markov chains ......... 52
D The Perron-Frobenius theorem .................................... 57
E The convergence theorem for positive recurrent Markov chains ... 63
F The ergodic theorem for positive recurrent Markov chains ...... 68
G p-recurrence . . .... ... .... ..... . ..... ...... 74
4 Reversible Markov chains 78
A The network model....................................... ... 78
B Speed of convergence of finite reversible Markov chains ...... 83
C The Poincare inequality........................................... 93
D Recurrence of infinite networks........... 102
E Random walks on integer lattices............................... . 109
5 Models of population evolution 116
A Birth-and-death Markov chains........... . ..................116
B The Galton-Watson process ................................ 131
C Branching Markov chains ....................... 140
viii Contents
6 Elements of the potential theory of transient Markov chains 153
A Motivation. The finite case .................................. 153
B Harmonic and superharmonic functions. Invariant and excessive
measures .................................................... 158
C Induced Markov chains.......................................... 164
D Potentials, Riesz decomposition, approximation ................ 169
E “Balayage” and domination principle.......... . ...............173
7 The Martin boundary of transient Markov chains 179
A Minimal harmonic functions ......................................179
B The Martin compactification..................................... 184
C Supermartingales, superharmonic functions, and excessive measures 191
D The Poisson-Martin integral representation theorem . ............200
E Poisson boundary. Alternative approach to the integral representation 209
8 Minimal harmonic functions on Euclidean lattices 219
9 Nearest neighbour random walks on trees 226
A Basic facts and computations................................... 226
B The geometric boundary of an infinite tree . ... . . ... ..... 232
C Convergence to ends and identification of the Martin boundary ... 237
D The integral representation of all harmonic functions . ... . . . . 246
E Limits of harmonic functions at the boundary . . . ........... . 251
F The boundary process, and the deviation from the limit geodesic . . 263
G Some recurrence / transience criteria ......................... 267
H Rate of escape and spectral radius .................. 279
Solutions of all exercises 297
Bibliography 339
A Textbooks and other general references.......................... 339
B Research-specific references................................... 341
List of symbols and notation 345
Index 349
|
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| indexdate | 2024-07-09T21:53:19Z |
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| isbn | 9783037190715 303719071X |
| language | English Italian |
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| physical | XVII, 351 S. graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Europ. Math. Soc. |
| record_format | marc |
| series2 | EMS textbooks in mathematics |
| spelling | Woess, Wolfgang 1954- Verfasser (DE-588)124183077 aut Denumerable Markov chains generating functions, boundary theory, random walks on trees Wolfgang Woess Zürich Europ. Math. Soc. 2009 XVII, 351 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier EMS textbooks in mathematics Boundary value problems Generating functions Markov processes Measure theory Random walks (Mathematics) Markov-Kette (DE-588)4037612-6 gnd rswk-swf Potenzialtheorie (DE-588)4046939-6 gnd rswk-swf Martin-Rand (DE-588)4451181-4 gnd rswk-swf Baum Mathematik (DE-588)4004849-4 gnd rswk-swf Irrfahrtsproblem (DE-588)4162442-7 gnd rswk-swf Markov-Kette (DE-588)4037612-6 s Potenzialtheorie (DE-588)4046939-6 s Martin-Rand (DE-588)4451181-4 s DE-604 Irrfahrtsproblem (DE-588)4162442-7 s Baum Mathematik (DE-588)4004849-4 s Erscheint auch als Woess, Wolfgang Denumerable Markov chains Online-Ausgabe 978-3-03719-571-0 (DE-604)BV036713275 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=018012749&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Woess, Wolfgang 1954- Denumerable Markov chains generating functions, boundary theory, random walks on trees Boundary value problems Generating functions Markov processes Measure theory Random walks (Mathematics) Markov-Kette (DE-588)4037612-6 gnd Potenzialtheorie (DE-588)4046939-6 gnd Martin-Rand (DE-588)4451181-4 gnd Baum Mathematik (DE-588)4004849-4 gnd Irrfahrtsproblem (DE-588)4162442-7 gnd |
| subject_GND | (DE-588)4037612-6 (DE-588)4046939-6 (DE-588)4451181-4 (DE-588)4004849-4 (DE-588)4162442-7 |
| title | Denumerable Markov chains generating functions, boundary theory, random walks on trees |
| title_auth | Denumerable Markov chains generating functions, boundary theory, random walks on trees |
| title_exact_search | Denumerable Markov chains generating functions, boundary theory, random walks on trees |
| title_full | Denumerable Markov chains generating functions, boundary theory, random walks on trees Wolfgang Woess |
| title_fullStr | Denumerable Markov chains generating functions, boundary theory, random walks on trees Wolfgang Woess |
| title_full_unstemmed | Denumerable Markov chains generating functions, boundary theory, random walks on trees Wolfgang Woess |
| title_short | Denumerable Markov chains |
| title_sort | denumerable markov chains generating functions boundary theory random walks on trees |
| title_sub | generating functions, boundary theory, random walks on trees |
| topic | Boundary value problems Generating functions Markov processes Measure theory Random walks (Mathematics) Markov-Kette (DE-588)4037612-6 gnd Potenzialtheorie (DE-588)4046939-6 gnd Martin-Rand (DE-588)4451181-4 gnd Baum Mathematik (DE-588)4004849-4 gnd Irrfahrtsproblem (DE-588)4162442-7 gnd |
| topic_facet | Boundary value problems Generating functions Markov processes Measure theory Random walks (Mathematics) Markov-Kette Potenzialtheorie Martin-Rand Baum Mathematik Irrfahrtsproblem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018012749&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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