Random Walks in the Quarter-Plane

Algebraic Methods, Boundary Value Problems and Applications

Guy Fayolle, Roudolf Iasnogorodski, Vadim Malyshev (Autoren)

Buch | Softcover

XV, 156 Seiten

2011 | 1. Softcover reprint of the original 1st ed. 1999
Springer Berlin (Verlag)
978-3-642-64217-3 (ISBN)

Artikel merken

Historical Comments Two-dimensional random walks in domains with non-smooth boundaries inter est several groups of the mathematical community. In fact these objects are encountered in pure probabilistic problems, as well as in applications involv ing queueing theory. This monograph aims at promoting original mathematical methods to determine the invariant measure of such processes. Moreover, as it will emerge later, these methods can also be employed to characterize the transient behavior. It is worth to place our work in its historical context. This book has three sources. l. Boundary value problems for functions of one complex variable; 2. Singular integral equations, Wiener-Hopf equations, Toeplitz operators; 3. Random walks on a half-line and related queueing problems. The first two topics were for a long time in the center of interest of many well known mathematicians: Riemann, Sokhotski, Hilbert, Plemelj, Carleman, Wiener, Hopf. This one-dimensional theory took its final form in the works of Krein, Muskhelishvili, Gakhov, Gokhberg, etc. The third point, and the related probabilistic problems, have been thoroughly investigated by Spitzer, Feller, Baxter, Borovkov, Cohen, etc.

and History.- 1 Probabilistic Background.- 1.1 Markov Chains.- 1.2 Random Walks in a Quarter Plane.- 1.3 Functional Equations for the Invariant Measure.- 2 Foundations of the Analytic Approach.- 2.1 Fundamental Notions and Definitions.- 2.2 Restricting the Equation to an Algebraic Curve.- 2.3 The Algebraic Curve Q(x, y) = 0.- 2.4 Galois Automorphisms and the Group of the Random Walk.- 2.5 Reduction of the Main Equation to the Riemann Torus.- 3 Analytic Continuation of the Unknown Functions in the Genus 1 Case.- 3.1 Lifting the Fundamental Equation onto the Universal Covering.- 3.2 Analytic Continuation.- 3.3 More about Uniformization.- 4 The Case of a Finite Group.- 4.1On the Conditions for H to be Finite.- 4.2 Rational Solutions.- 4.3 Algebraic Solution.- 4.4 Final Form of the General Solution.- 4.5 The Problem of the Poles and Examples.- 4.6 An Example of Algebraic Solution by Flatto and Hahn.- 4.7 Two Queues in Tandem.- 5 Solution in the Case of an Arbitrary Group.- 5.1 Informal Reduction to a Riemann-Hilbert-Carleman BVP.- 5.2 Introduction to BVP in the Complex Plane.- 5.3 Further Properties of the Branches Defined by Q(x, y)= 0.- 5.4 Index and Solution of the BVP (5.1.5).- 5.5 Complements.- 6 The Genus 0 Case.- 6.1 Properties of the Branches.- 6.2 Case 1: ?01 = ??1,0 = ??1,1 = 0.- 6.3 Case 3: ?11 = ?10 = ?01 = 0.- 6.4 Case 4: ??1,0 = ?0,?1 = ??1,?1= 0.- 6.5 Case 5: MZ= My= 0.- 7 Miscellanea.- 7.1 About Explicit Solutions.- 7.2 Asymptotics.- 7.3 Generalized Problems and Analytic Continuation.- 7.4 Outside Probability.- References.

Erscheint lt. Verlag	18.9.2011
Reihe/Serie	Stochastic Modelling and Applied Probability
Zusatzinfo	XV, 156 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	276 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
Themenwelt	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
Schlagworte	Analysis • functional equations • Markov Chain • measure • queuing theory • Random Walk • random walks • Riemann Surfaces • uniformization
ISBN-10	3-642-64217-9 / 3642642179
ISBN-13	978-3-642-64217-3 / 9783642642173
Zustand	Neuware