Elements of Random Walk and Diffusion Processes:
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| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York, NY
Wiley
2013
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| Schriftenreihe: | Wiley Series in Operations Research and Management Science
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XV, 260 S. graph. Darst. |
| ISBN: | 9781118618097 |
Internformat
MARC
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| 245 | 1 | 0 | |a Elements of Random Walk and Diffusion Processes |c Oliver C. Ibe |
| 264 | 1 | |a New York, NY |b Wiley |c 2013 | |
| 300 | |a XV, 260 S. |b graph. Darst. | ||
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Datensatz im Suchindex
| _version_ | 1804151551342673920 |
|---|---|
| adam_text | CONTENTS
PREFACE
xiii
ACKNOWLEDGMENTS
xv
1
REVIEW OF PROBABILITY THEORY
1
1.1
Introduction
1
1.2
Random Variables
1
1.2.1
Distribution Functions
2
1.2.2
Discrete Random Variables
3
1.2.3
Continuous Random Variables
3
1.2.4
Expectations
4
1.2.5
Moments of Random Variables and the Variance
4
1.3
Transform Methods
5
1.3.1
The Characteristic Function
5
1.3.2
Moment-Generating Property of the Characteristic
Function
6
1.3.3
The s-Transform
6
1.3.4
Moment-Generating Property of the s-Transform
7
1.3.5
The z-Transform
7
1.3.6
Moment-Generating Property of the z-Transform
8
1.4
Covariance and Correlation Coefficient
9
VÍ
CONTENTS
1.5
Sums of Independent Random Variables
10
1.6
Some Probability Distributions
11
1.6.1
The Bernoulli Distribution
11
1.6.2
The Binomial Distribution
12
1.6.3
The Geometric Distribution
12
1.6.4
The
Poisson
Distribution
13
1.6.5
The Exponential Distribution
13
1.6.6
Normal Distribution
14
1.7
Limit Theorems
16
1.7.1
Markov Inequality
16
1.7.2
Chebyshev Inequality
17
1.7.3
Laws of Large Numbers
17
1.7.4
The Central Limit Theorem
18
Problems
19
2
OVERVIEW OF STOCHASTIC PROCESSES
21
2.1
Introduction
21
2.2
Classification of Stochastic Processes
22
2.3
Mean and Autocorrelation Function
22
2.4
Stationary Processes
23
2.4.1
Strict-Sense Stationary Processes
23
2.4.2
Wide-Sense Stationary Processes
24
2.5
Power Spectral Density
24
2.6
Counting Processes
25
2.7
Independent Increment Processes
25
2.8
Stationary Increment Process
25
2.9
Poisson
Processes
26
2.9.1
Compound
Poisson
Process
28
2.10
Markov Processes
29
2.10.1
Discrete-Time Markov Chains
30
2.10.2
State Transition Probability Matrix
31
2.10.3
The ¿-Step State Transition Probability
31
2.10.4
State Transition Diagrams
32
2.10.5
Classification of States
33
2.10.6
Limiting-State Probabilities
34
2.10.7
Doubly Stochastic Matrix
35
2.10.8
Continuous-Time Markov Chains
35
2.10.9
Birth and Death Processes
36
CONTENTS
VM
2.11
Gaussian
Processes
38
2.12
Martingales
38
2.12.1
Stopping Times
40
Problems
41
ONE-DIMENSIONAL RANDOM WALK
44
3.1
Introduction
44
3.2
Occupancy Probability
46
3.3
Random Walk as a Markov Chain
49
3.4
Symmetric Random Walk as a Martingale
49
3.5
Random Walk with Barriers
50
3.6
Mean-Square Displacement
50
3.7
Gambler s Ruin
52
3.7.1
Ruin Probability
52
3.7.2
Alternative Derivation of Ruin Probability
54
3.7.3
Duration of a Game
55
3.8
Random Walk with Stay
56
3.9
First Return to the Origin
57
3.10
First Passage Times for Symmetric Random Walk
59
3.10.1
First Passage Time via the Generating Function
59
3.10.2
First Passage Time via the Reflection Principle
61
3.10.3
Hitting Time and the Reflection Principle
64
3.11
The Ballot Problem and the Reflection Principle
65
3.11.1
The Conditional Probability Method
66
3.12
Returns to the Origin and the Arc-Sine Law
67
3.13
Maximum of a Random Walk
72
3.14
Two Symmetric Random Walkers
73
3.15
Random Walk on a Graph
73
3.15.1
Proximity Measures
75
3.15.2
Directed Graphs
75
3.15.3
Random Walk on an Undirected Graph
76
3.15.4
Random Walk on a Weighted Graph
80
3.16
Random Walks and Electric Networks
80
3.16.1
Harmonic Functions
82
3.16.2
Effective Resistance and Escape Probability
82
3.17
Correlated Random Walk
85
3.18
Continuous-Time Random Walk
90
3.18.1
The Master Equation
92
VIII CONTENTS
3.19
Reinforced Random Walk
94
3.19.1
Polya s Urn Model
94
3.19.2
ERRW and Polya s Urn
96
3.19.3
ERRW Revisited
97
3.20
Miscellaneous Random Walk Models
98
3.20.1
Geometric Random Walk
98
3.20.2
Gaussian Random Walk
99
3.20.3
Random Walk with Memory
99
3.21
Summary
100
Problems
100
4
TWO-DIMENSIONAL RANDOM WALK
103
4.
1 Introduction
103
4.2
The Pearson Random Walk
105
4.2.1
Mean-Square Displacement
105
4.2.2
Probability Distribution
107
4.3
The Symmetric 2D Random Walk
110
4.3.1
Stirling s Approximation of Symmetric
Walk
112
4.3.2
Probability of Eventual Return for Symmetric
Walk
113
4.3.3
Mean-Square Displacement
114
4.3.4
Two Independent Symmetric 2D Random
Walkers
114
4.4
The Alternating Random Walk
115
4.4.1
Stirling s Approximation of Alternating
Walk
117
4.4.2
Probability of Eventual Return for Alternating
Walk
117
4.5
Self-Avoiding Random Walk
117
4.6
Nonreversing Random Walk
121
4.7
Extensions of the NRRW
126
4.7.1
The Noncontinuing Random Walk
126
4.7.2
The Nonreversing and Noncontinuing Random
Walk
127
4.8
Summary
128
5
BROWNIAN MOTION
129
5.1
Introduction
129
5.2
Brownian Motion with Drift
132
CONTENTS
ІХ
5.3 Brownian Motion
as a Markov Process
132
5.4
Brownian Motion as a Martingale
133
5.5
First Passage Time of a Brownian Motion
133
5.6
Maximum of a Brownian Motion
135
5.7
First Passage Time in an Interval
135
5.8
The Brownian Bridge
136
5.9
Geometric Brownian Motion
137
5.10
The
Langevin
Equation
137
5.11
Summary
141
Problems
141
INTRODUCTION TO STOCHASTIC CALCULUS
143
6.1
Introduction
143
6.2
The
Ito
Integral
145
6.3
The Stochastic Differential
146
6.4
The Ito s Formula
147
6.5
Stochastic Differential Equations
147
6.6
Solution of the Geometric Brownian Motion
148
6.7
The Ornstein-Uhlenbeck Process
151
6.7.1
Solution of the Ornstein-Uhlenbeck SDE
152
6.7.2
First Alternative Solution Method
153
6.7.3
Second Alternative Solution Method
154
6.8
Mean-Reverting Ornstein—Uhlenbeck Process
155
6.9
Summary
157
DIFFUSION PROCESSES
158
7.1
Introduction
158
7.2
Mathematical Preliminaries
159
7.3
Diffusion on One-Dimensional Random Walk
160
7.3.1
Alternative Derivation
163
7.4
Examples of Diffusion Processes
164
7.4.1
Brownian Motion
164
7.4.2
Brownian Motion with Drift
167
7.5
Correlated Random Walk and the Telegraph Equation
167
7.6
Diffusion at Finite Speed
170
7.7
Diffusion on Symmetric Two-Dimensional Lattice
Random Walk
171
7.8
Diffusion Approximation of the Pearson Random Walk
173
7.9
Summary
174
X
CONTENTS
8
LEVY WALK
175
8.
1 Introduction
175
8.2
Generalized Central Limit Theorem
175
8.3
Stable Distribution
177
8.4
Self-Similarity
182
8.5
Fractals
183
8.6
Levy Distribution
185
8.7
Levy Process
186
8.8
Infinite Divisibility
186
8.8.1
The Infinite Divisibility of the
Poisson
Process
187
8.8.2
Infinite Divisibility of the Compound
Poisson
Process
187
8.8.3
Infinite Divisibility of the Brownian
Motion with Drift
188
8.9
Levy Flight
188
8.9.1
First Passage Time of Levy Flights
190
8.9.2
Leapover Properties of Levy Flights
190
8.10
Truncated Levy Flight
191
8.11
Levy Walk
191
8.11.1
Levy Walk as a Coupled CTRW
192
8.11.2
Truncated Levy Walk
195
8.12
Summary
195
9
FRACTIONAL CALCULUS AND ITS APPLICATIONS
196
9.1
Introduction
196
9.2
Gamma Function
197
9.3
Mittag-Leffler Functions
198
9.4
Laplace Transform
200
9.5
Fractional Derivatives
202
9.6
Fractional Integrals
203
9.7
Definitions of Fractional Integro-Differentials
203
9.7.1
Riemann—Liouville Fractional Derivative
204
9.7.2
Caputo
Fractional Derivative
205
9.7.3
Granwald-Letnikov Fractional Derivative
206
9.8
Fractional Differential Equations
207
9.8.1
Relaxation Differential Equation of Integer Order
208
9.8.2
Oscillation Differential Equation of Integer
Order
208
9.8.3
Relaxation and Oscillation Fractional Differential
Equations
209
CONTENTS
ХІ
9.9 Applications
of Fractional Calculus
210
9.9.1
Fractional Brownian Motion
210
9.9.2
Multifractional Brownian Motion
213
9.9.3
Fractional Random Walk
213
9.9.4
Fractional (or Anomalous) Diffusion
215
9.9.5
Fractional Gaussian Noise
221
9.9.6
Fractional
Poisson
Process
222
9.10
Summary
224
10
PERCOLATION THEORY
225
10.1
Introduction
225
10.2
Graph Theory Revisited
226
10.2.1
Complete Graphs
226
10.2.2
Random Graphs
226
10.3
Percolation on a Lattice
228
10.3.1
Cluster Formation and Phase Transition
229
10.3.2
Percolation Probability and Critical Exponents
233
10.4
Continuum Percolation
235
10.4.1
The Boolean Model
235
10.4.2
The Random Connection Model
236
10.5
Bootstrap (or fc-Core) Percolation
237
10.6
Diffusion Percolation
237
10.6.1
Bootstrap Percolation versus Diffusion Percolation
239
10.7
First-Passage Percolation
239
10.8
Explosive Percolation
240
10.9
Percolation in Complex Networks
242
10.9.1
Average Path Length
243
10.9.2
Clustering Coefficient
243
10.9.3
Degree Distribution
244
10.9.4
Percolation and Network Resilience
244
10.10
Summary
245
REFERENCES
247
INDEX
253
|
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| id | DE-604.BV041435418 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:56:37Z |
| institution | BVB |
| isbn | 9781118618097 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-026882296 |
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| physical | XV, 260 S. graph. Darst. |
| publishDate | 2013 |
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| record_format | marc |
| series2 | Wiley Series in Operations Research and Management Science |
| spelling | Ibe, Oliver C. 1947- Verfasser (DE-588)136641784 aut Elements of Random Walk and Diffusion Processes Oliver C. Ibe New York, NY Wiley 2013 XV, 260 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Wiley Series in Operations Research and Management Science Stochastik (DE-588)4121729-9 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Zufallsauswahl (DE-588)4191095-3 gnd rswk-swf Stochastik (DE-588)4121729-9 s Wahrscheinlichkeitstheorie (DE-588)4079013-7 s Zufallsauswahl (DE-588)4191095-3 s DE-604 Digitalisierung UB Bamberg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026882296&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ibe, Oliver C. 1947- Elements of Random Walk and Diffusion Processes Stochastik (DE-588)4121729-9 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Zufallsauswahl (DE-588)4191095-3 gnd |
| subject_GND | (DE-588)4121729-9 (DE-588)4079013-7 (DE-588)4191095-3 |
| title | Elements of Random Walk and Diffusion Processes |
| title_auth | Elements of Random Walk and Diffusion Processes |
| title_exact_search | Elements of Random Walk and Diffusion Processes |
| title_full | Elements of Random Walk and Diffusion Processes Oliver C. Ibe |
| title_fullStr | Elements of Random Walk and Diffusion Processes Oliver C. Ibe |
| title_full_unstemmed | Elements of Random Walk and Diffusion Processes Oliver C. Ibe |
| title_short | Elements of Random Walk and Diffusion Processes |
| title_sort | elements of random walk and diffusion processes |
| topic | Stochastik (DE-588)4121729-9 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Zufallsauswahl (DE-588)4191095-3 gnd |
| topic_facet | Stochastik Wahrscheinlichkeitstheorie Zufallsauswahl |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026882296&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT ibeoliverc elementsofrandomwalkanddiffusionprocesses |