| 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： |
図書
