Random walks in the quarter plane: algebraic methods, boundary value problems and applications
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1999
|
| Series: | Applications of mathematics
40 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 156 S. graph. Darst. |
| ISBN: | 3540650474 |
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MARC
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| 245 | 1 | 0 | |a Random walks in the quarter plane |b algebraic methods, boundary value problems and applications |c Guy Fayolle ; Roudolf Iasnogorodski ; Vadim Malyshev |
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Record in the Search Index
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| adam_text | Table of Contents
Introduction and History VII
1 Probabilistic Background 1
1.1 Markov Chains 1
1.2 Random Walks in a Quarter Plane 2
1.3 Functional Equations for the Invariant Measure 3
2 Foundations of the Analytic Approach 7
2.1 Fundamental Notions and Definitions 7
2.1.1 Covering Manifolds 8
2.1.2 Algebraic Functions 10
2.1.3 Elements of Galois Theory 11
2.1.4 Universal Cover and Uniformization 13
2.1.5 Abelian Differentials and Divisors 13
2.2 Restricting the Equation to an Algebraic Curve 14
2.2.1 First Insight (Algebraic Functions) 15
2.2.2 Second Insight (Algebraic Curve) 15
2.2.3 Third Insight (Factorization) 16
2.2.4 Fourth Insight (Riemann Surfaces) 17
2.3 The Algebraic Curve Q(x, y) = 0 18
2.3.1 Branches of the Algebraic Functions on the Unit Circle . 21
2.3.2 Branch Points 23
2.4 Galois Automorphisms and the Group of the Random Walk . . 27
2.4.1 Construction of the Automorphisms £ and fj on S .... 29
2.5 Reduction of the Main Equation to the Riemann Torus 30
3 Analytic Continuation of the Unknown Functions in the Genus
1 Case 35
3.1 Lifting the Fundamental Equation onto the Universal Covering 35
3.1.1 Lifting of the Branch Points 37
3.1.2 Lifting of the Automorphisms on the Universal Covering 37
3.2 Analytic Continuation 39
3.3 More about Uniformization 42
XIV Table of Contents
4 The Case of a Finite Group 51
4.1 On the Conditions for H. to be Finite 51
4.1.1 Explicit Conditions for Groups of Order 4 or 6 52
4.1.2 The General Case 54
4.2 Rational Solutions 56
4.2.1 The Case N(f) ^ 1 58
4.2.2 The Case N(f) = 1 59
4.3 Algebraic Solutions 67
4.3.1 The Case N(f) = 1 67
4.3.2 The Case N(f) ^ 1 72
4.4 Final Form of the General Solution 74
4.5 The Problem of the Poles and Examples 79
4.5.1 Rational Solutions 79
4.5.1.1 Reversible Random Walks 79
4.5.1.2 Simple Examples of Nonreversible Random Walks 79
4.5.1.3 One Parameter Families 83
4.5.1.4 Two Typical Situations 83
4.5.1.5 Ergodicity Conditions 85
4.5.1.6 Proof of Lemma 4.5.2 86
4.6 An Example of Algebraic Solution by Flatto and Hahn 88
4.7 Two Queues in Tandem 91
5 Solution in the Case of an Arbitrary Group 93
5.1 Informal Reduction to a Riemann Hilbert Carleman BVP .... 93
5.2 Introduction to BVP in the Complex Plane 95
5.2.1 A Bit of History 95
5.2.2 The Sokhotski Plemelj Formulae 96
5.2.3 The Riemann Boundary Value Problem for a Closed Con¬
tour 97
5.2.4 The Riemann BVP for an Open Contour 100
5.2.5 The Riemann Carleman Problem with a Shift 102
5.3 Further Properties of the Branches Denned by Q(x,y) = 0 . . . 109
5.4 Index and Solution of the BVP (5.1.5) 119
5.5 Complements 125
5.5.1 Analytic Continuation 125
5.5.2 Computation of w 125
5.5.2.1 An Explicit Form via the Weierstrass p Function. 125
5.5.2.2 A Differential Equation 127
5.5.2.3 An Integral Equation 127
6 The Genus 0 Case 129
6.1 Properties of the Branches 129
6.2 Case 1: pOi = P i,o = P i,i =0 131
6.3 Case 3: pn = pm = pOi = 0 132
6.4 Case 4: p_ii0 = po, i = P i, i =0 134
Table of Contents XV
6.4.1 Integral Equation 135
6.4.2 Series Representation 135
6.4.3 Uniformization 135
6.4.4 Boundary Value Problem 137
6.5 Case 5: Mx = My = 0 137
7 Miscellanea 145
7.1 About Explicit Solutions 145
7.2 Asymptotics 145
7.2.1 Large Deviations and Stationary Probabilities 146
7.3 Generalized Problems and Analytic Continuation 148
7.4 Outside Probability 150
References 151
Index 155
|
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| isbn | 3540650474 |
| language | English |
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| spelling | Fayolle, Guy 1943- Verfasser (DE-588)120682036 aut Random walks in the quarter plane algebraic methods, boundary value problems and applications Guy Fayolle ; Roudolf Iasnogorodski ; Vadim Malyshev Berlin [u.a.] Springer 1999 XV, 156 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applications of mathematics 40 Promenades aléatoires (Mathématiques) ram Random walks (statistiek) gtt Random walks (Mathematics) Viertelebene (DE-588)4553601-6 gnd rswk-swf Irrfahrtsproblem (DE-588)4162442-7 gnd rswk-swf Irrfahrtsproblem (DE-588)4162442-7 s Viertelebene (DE-588)4553601-6 s DE-604 Iasnogorodski, Roudolf 1938- Verfasser (DE-588)120682060 aut Malyshev, Vadim 1938- Verfasser (DE-588)120682079 aut Applications of mathematics 40 (DE-604)BV000895226 40 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008349641&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fayolle, Guy 1943- Iasnogorodski, Roudolf 1938- Malyshev, Vadim 1938- Random walks in the quarter plane algebraic methods, boundary value problems and applications Applications of mathematics Promenades aléatoires (Mathématiques) ram Random walks (statistiek) gtt Random walks (Mathematics) Viertelebene (DE-588)4553601-6 gnd Irrfahrtsproblem (DE-588)4162442-7 gnd |
| subject_GND | (DE-588)4553601-6 (DE-588)4162442-7 |
| title | Random walks in the quarter plane algebraic methods, boundary value problems and applications |
| title_auth | Random walks in the quarter plane algebraic methods, boundary value problems and applications |
| title_exact_search | Random walks in the quarter plane algebraic methods, boundary value problems and applications |
| title_full | Random walks in the quarter plane algebraic methods, boundary value problems and applications Guy Fayolle ; Roudolf Iasnogorodski ; Vadim Malyshev |
| title_fullStr | Random walks in the quarter plane algebraic methods, boundary value problems and applications Guy Fayolle ; Roudolf Iasnogorodski ; Vadim Malyshev |
| title_full_unstemmed | Random walks in the quarter plane algebraic methods, boundary value problems and applications Guy Fayolle ; Roudolf Iasnogorodski ; Vadim Malyshev |
| title_short | Random walks in the quarter plane |
| title_sort | random walks in the quarter plane algebraic methods boundary value problems and applications |
| title_sub | algebraic methods, boundary value problems and applications |
| topic | Promenades aléatoires (Mathématiques) ram Random walks (statistiek) gtt Random walks (Mathematics) Viertelebene (DE-588)4553601-6 gnd Irrfahrtsproblem (DE-588)4162442-7 gnd |
| topic_facet | Promenades aléatoires (Mathématiques) Random walks (statistiek) Random walks (Mathematics) Viertelebene Irrfahrtsproblem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008349641&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000895226 |
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