Ebook: Random Walks in the Quarter-Plane: Algebraic Methods, Boundary Value Problems and Applications

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.

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This monograph aims at promoting original mathematical methods to determine the invariant measure of two-dimensional random walks in domains with boundaries. Such processes are of interest in several areas of mathematical research and are encountered in pure probabilistic problems, as well as in applications involving queuing theory. Using Riemann surfaces and boundary value problems, the authors propose completely new approaches to solve functional equations of two complex variables. These methods can also be employed to characterize the transient behavior of random walks in the quarter plane.

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This monograph aims at promoting original mathematical methods to determine the invariant measure of two-dimensional random walks in domains with boundaries. Such processes are of interest in several areas of mathematical research and are encountered in pure probabilistic problems, as well as in applications involving queuing theory. Using Riemann surfaces and boundary value problems, the authors propose completely new approaches to solve functional equations of two complex variables. These methods can also be employed to characterize the transient behavior of random walks in the quarter plane.
Content:
Front Matter....Pages I-XV
Probabilistic Background....Pages 1-6
Foundations of the Analytic Approach....Pages 7-33
Analytic Continuation of the Unknown Functions in the Genus 1 Case....Pages 35-49
The Case of a Finite Group....Pages 51-91
Solution in the Case of an Arbitrary Group....Pages 93-127
The Genus 0 Case....Pages 129-143
Miscellanea....Pages 145-150
Erratum....Pages E1-E4
Back Matter....Pages 151-154

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This monograph aims at promoting original mathematical methods to determine the invariant measure of two-dimensional random walks in domains with boundaries. Such processes are of interest in several areas of mathematical research and are encountered in pure probabilistic problems, as well as in applications involving queuing theory. Using Riemann surfaces and boundary value problems, the authors propose completely new approaches to solve functional equations of two complex variables. These methods can also be employed to characterize the transient behavior of random walks in the quarter plane.
Content:
Front Matter....Pages I-XV
Probabilistic Background....Pages 1-6
Foundations of the Analytic Approach....Pages 7-33
Analytic Continuation of the Unknown Functions in the Genus 1 Case....Pages 35-49
The Case of a Finite Group....Pages 51-91
Solution in the Case of an Arbitrary Group....Pages 93-127
The Genus 0 Case....Pages 129-143
Miscellanea....Pages 145-150
Erratum....Pages E1-E4
Back Matter....Pages 151-154
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