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: |