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＜電子ブック＞
Random Walks in the Quarter-Plane : Algebraic Methods, Boundary Value Problems and Applications / by Guy Fayolle, Roudolf Iasnogorodski, Vadim Malyshev
(Stochastic Modelling and Applied Probability. ISSN:2197439X ; 40)

版	1st ed. 1999.
出版者	(Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer)
出版年	1999
本文言語	英語
大きさ	XV, 156 p : online resource
著者標目	*Fayolle, Guy author Iasnogorodski, Roudolf author Malyshev, Vadim author SpringerLink (Online service)
件　名	LCSH:Probabilities LCSH:Mathematical analysis LCSH:Statistics  FREE:Probability Theory FREE:Analysis FREE:Statistical Theory and Methods
一般注記	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 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 HTTP:URL=https://doi.org/10.1007/978-3-642-60001-2

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