Random walk: a modern introduction
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
[2010]
|
| Edition: | First published |
| Series: | Cambridge studies in advanced mathematics
123 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xii, 364 Seiten Illustrationen |
| ISBN: | 9780521519182 |
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MARC
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| 245 | 1 | 0 | |a Random walk |b a modern introduction |c Gregory F. Lawler ; Vlada Limic |
| 250 | |a First published | ||
| 264 | 1 | |a Cambridge |b Cambridge University Press |c [2010] | |
| 264 | 4 | |c © 2010 | |
| 300 | |a xii, 364 Seiten |b Illustrationen | ||
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| 338 | |b nc |2 rdacarrier | ||
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| 830 | 0 | |a Cambridge studies in advanced mathematics |v 123 |w (DE-604)BV000003678 |9 123 | |
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Record in the Search Index
| _version_ | 1810097222397722624 |
|---|---|
| adam_text |
Contents
Preface
page
ix
Introduction
1
1.1
Basic definitions
1
.2
Continuous-time random walk
6
.3
Other lattices
7
.4
Other walks
11
.5
Generator
11
.6
Filtrations and strong Markov property
14
.7
A word about constants
17
Local centra] limit theorem
21
2.1
Introduction
21
2.2
Characteristic functions and LCLT
25
2.2.1
Characteristic functions of random variables in M.d
25
2.2.2
Characteristic functions of random variables in Zd
27
2.3
LCLT
-
characteristic function approach
28
2.3.1
Exponential moments
46
2.4
Some corollaries of the LCLT
51
2.5
LCLT
-
combinatorial approach
58
2.5.1
Stirling's formula and one-dimensional walks
58
2.5.2
LCLT for
Poisson
and continuous-time walks
64
Approximation by Brow
nian
motion
72
3.1
Introduction
72
3.2
Construction of Brownian motion
74
3.3
Skorokhod embedding
79
3.4
Higher dimensions
82
3.5
An alternative formulation
84
vi
Contents
4
The Green's function
87
4.1
Recurrence and transience
87
4.2
The Green's generating function
88
4.3
The Green's function, transient case
95
4.3.1
Asymptotics under weaker assumptions
99
4.4
Potential kernel
101
4.4.1
Two dimensions
101
4.4.2
Asymptotics under weaker assumptions
107
4.4.3
One dimension
109
4.5
Fundamental solutions
113
4.6
The Green's function for a set
114
5
One-dimensional walks
123
5.1
Gambler's ruin estimate
123
5.1.1
General case
127
5.2
One-dimensional killed walks
135
5.3
Hitting a half-line
138
6
Potential theory
144
6.1
Introduction
144
6.2
Dirichlet problem
146
6.3
Difference estimates and Harnack inequality
152
6.4
Further estimates
160
6.5
Capacity, transient case
166
6.6
Capacity in two dimensions
176
6.7
Neumann problem
186
6.8
Beurling estimate
189
6.9
Eigenvalue of a set
194
7
Dyadic coupling
205
7.1
Introduction
205
7.2
Some estimates
207
7.3
Quantile coupling
210
7.4
The dyadic coupling
213
7.5
Proof of Theorem
7.1.1 216
7.6
Higher dimensions
218
7.7
Coupling the exit distributions
219
8
Additional topics on simple random walk
225
8.1
Poisson
kernel
225
8.1.1
Half space
226
Contents
vii
8.1.2
Cube
229
8.1.3 Strips
and quadrants in Z2
235
8.2
Eigenvalues for rectangles
238
8.3
Approximating continuous harmonic functions
239
8.4
Estimates for the ball
241
9
Loop measures
247
9.1
Introduction
247
9.2
Definitions and notations
247
9.2.1
Simple random walk on a graph
251
9.3
Generating functions and loop measures
252
9.4
Loop soup
257
9.5
Loop erasure
259
9.6
Boundary excursions
261
9.7
Wilson's algorithm and spanning trees
268
9.8
Examples
271
9.8.1
Complete graph
271
9.8.2
Hypercube
272
9.8.3
Sierpinski graphs
275
9.9
Spanning trees of subsets of Z2
277
9.10
Gaussian free field
289
10
Intersection probabilities for random walks
297
10.1
Long-range estimate
297
10.2
Short-range estimate
302
10.3
One-sided exponent
305
11
Loop-erased random walk
307
11.1
A-processes
307
11.2
Loop-erased random walk
311
11.3
LERWinZ^
313
11.3.1
d >3
314
11.3.2
d
=2 315
11.4
Rate of growth
319
11.5
Short-range intersections
323
Appendix
326
A.
1
Some expansions
326
A.
1.1
Riemann sums
326
A.
1.2
Logarithm
327
A.2 Martingales
331
A.2.
1
Optional sampling theorem
332
viii Contents
А.З
A.4
A.5
A.6
A.7
Bibliography
Index
с
Index
A.I.I Maximal inequality
334
A.
2.3
Continuous martingales
336
Joint normal distributions
337
Markov chains
339
A.4.
1
Chains restricted to subsets
342
A.4.
2
Maximal coupling of Markov chains
346
Some Tauberian theory
351
Second moment method
353
Subadditivity
354
iv
360
nbols
361
363 |
| any_adam_object | 1 |
| author | Lawler, Gregory F. 1955- Limic, Vlada |
| author_GND | (DE-588)123908671 (DE-588)141817623 |
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| discipline | Mathematik |
| edition | First published |
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| id | DE-604.BV036561331 |
| illustrated | Illustrated |
| indexdate | 2024-09-13T16:00:31Z |
| institution | BVB |
| isbn | 9780521519182 |
| language | English |
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| physical | xii, 364 Seiten Illustrationen |
| publishDate | 2010 |
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| publisher | Cambridge University Press |
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| series | Cambridge studies in advanced mathematics |
| series2 | Cambridge studies in advanced mathematics |
| spelling | Lawler, Gregory F. 1955- (DE-588)123908671 aut Random walk a modern introduction Gregory F. Lawler ; Vlada Limic First published Cambridge Cambridge University Press [2010] © 2010 xii, 364 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 123 Irrfahrtsproblem (DE-588)4162442-7 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Irrfahrtsproblem (DE-588)4162442-7 s DE-604 Limic, Vlada (DE-588)141817623 aut Erscheint auch als Online-Ausgabe 978-0-511-75085-4 Cambridge studies in advanced mathematics 123 (DE-604)BV000003678 123 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020482662&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lawler, Gregory F. 1955- Limic, Vlada Random walk a modern introduction Cambridge studies in advanced mathematics Irrfahrtsproblem (DE-588)4162442-7 gnd |
| subject_GND | (DE-588)4162442-7 (DE-588)4123623-3 |
| title | Random walk a modern introduction |
| title_auth | Random walk a modern introduction |
| title_exact_search | Random walk a modern introduction |
| title_full | Random walk a modern introduction Gregory F. Lawler ; Vlada Limic |
| title_fullStr | Random walk a modern introduction Gregory F. Lawler ; Vlada Limic |
| title_full_unstemmed | Random walk a modern introduction Gregory F. Lawler ; Vlada Limic |
| title_short | Random walk |
| title_sort | random walk a modern introduction |
| title_sub | a modern introduction |
| topic | Irrfahrtsproblem (DE-588)4162442-7 gnd |
| topic_facet | Irrfahrtsproblem Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020482662&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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