Random walks in the quarter-plane : algebraic methods, boundary value problems and applications

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Random walks in the quarter-plane : algebraic methods, boundary value problems and applications / Guy Fayolle, Roudolf Iasnogorodski, Vadim Malyshev

資料種別:

図書

出版情報:

Berlin ; New York : Springer, c1999

形態:

xv, 156 p. ; 25 cm

シリーズ名:

Applications of mathematics ; 40 <BA00256802>

著者名:

ISBN:

9783540650478 [3540650474] (: hardcover)

書誌ID:

BA41448156

子書誌情報

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所蔵情報

他の版・巻

書誌詳細

ISSN:

01724568

注記:

Includes bibliographical references and index

主題:

Random walks (Mathematics)

言語:

英語

目次情報:

Introduction and History

Probabilistic Background / 1：

Markov Chains / 1.1：

Random Walks in a Quarter Plane / 1.2：

Functional Equations for the Invariant Measure / 1.3：

Foundations of the Analytic Approach / 2：

Fundamental Notions and Definitions / 2.1：

Covering Manifolds / 2.1.1：

Algebraic Functions / 2.1.2：

Elements of Galois Theory / 2.1.3：

Universal Cover and Uniformization / 2.1.4：

Abelian Differentials and Divisors / 2.1.5：

Restricting the Equation to an Algebraic Curve / 2.2：

First Insight (Algebraic Functions) / 2.2.1：

Second Insight (Algebraic Curve) / 2.2.2：

Third Insight (Factorization) / 2.2.3：

Fourth Insight (Riemann Surfaces) / 2.2.4：

The Algebraic Curve Q(x,y) = 0 / 2.3：

Branches of the Algebraic Functions on the Unit Circle / 2.3.1：

Branch Points / 2.3.2：

Galois Automorphisms and the Group of the Random Walk / 2.4：

? and ? on S / 2.4.1：

Reduction of the Main Equation to the Riemann Torus / 2.5：

Analytic Continuation of the Unknown Functions in the Genus Case / 3：

Lifting the Fundamental Equation onto the Universal Covering / 3.1：

Lifting of the Branch Points / 3.1.1：

Lifting of the Automorphisms on the Universal Covering / 3.1.2：

Analytic Continuation / 3.2：

More about Uniformization / 3.3：

The Case of a Finite Group / 4：

On the Conditions for H to be Finite / 4.1：

Explicit Conditions for Groups of Order 4 or 6 / 4.1.1：

The General Case / 4.1.2：

Rational Solutions / 4.2：

The Case N(f) = 1 / 4.2.1：

Algebraic Solutions / 4.2.2：

Final Form of the General Solution / 4.3.1：

The Problem of the Poles and Examples / 4.5：

Reversible Random Walks / 4.5.1：

Simple Examples of Nonreversible Random Walks / 4.5.1.2：

One Parameter Families / 4.5.1.3：

Two Typical Situations / 4.5.1.4：

Ergodicity Conditions / 4.5.1.5：

Proof of Lemma 4.5.2 / 4.5.1.6：

An Example of Algebraic Solution by Flatto and Hahn / 4.6：

Two Queues in Tandem / 4.7：

Solution in the Case of an Arbitrary Group / 5：

Informal Reduction to a Riemann-Hilbert-Carleman BVP / 5.1：

Introduction to BVP in the Complex Plane / 5.2：

A Bit of History / 5.2.1：

The Sokhotski-Plemelj Formulae / 5.2.2：

The Riemann Boundary Value Problem for a Closed Con- tour / 5.2.3：

The Riemann BVP for an Open Contour / 5.2.4：

The Riemann-Carleman Problem with a Shift / 5.2.5：

Further Properties of the Branches Defined by Q(x,y) = 0 / 5.3：

Index and Solution of the BVP (5.1.5) / 5.4：

Complements / 5.5：

Computation of w / 5.5.1：

An Explicit Form via the Weierstrass P-Function / 5.5.2.1：

A Differential Equation / 5.5.2.2：

An Integral Equation / 5.5.2.3：

The Genus 0 Case / 6：

Properties of the Branches / 6.1：

Case 1: <$$$> / 6.2：

Case 3: <$$$> / 6.3：

Case 4: <$$$> / 6.4：

Integral Equation / 6.4.1：

Series Representation / 6.4.2：

Uniformization / 6.4.3：

Boundary Value Problem / 6.4.4：

Case 5: <$$$> / 6.5：

Miscellanea / 7：

About Explicit Solutions / 7.1：

Asymptotics / 7.2：

Large Deviations and Stationary Probabilities / 7.2.1：

Generalized Problems and Analytic Continuation / 7.3：

Outside Probability / 7.4：

References

Index

Introduction and History

Probabilistic Background / 1：

Markov Chains / 1.1：

Random Walks in a Quarter Plane / 1.2：

Functional Equations for the Invariant Measure / 1.3：

Foundations of the Analytic Approach / 2：

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