Random Walks in the Quarter-Plane: Algebraic Methods, Boundary Value Problems and Applications
Springer Science & Business Media, May 4, 1999 - Mathematics - 156 pages
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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addition algebraic algebraic curve algebraic function analytic continuation apply arbitrary argument assume automorphisms belong boundary bounded branch points called chapter closed complex plane concluded condition connected Consequently consider contour corresponding curve defined definition denoted depend derivative differential direction domain elements elliptic equal equation equivalent ergodic exactly exists fact field figure finite formulae function fundamental Galois genus given gives Hence holds holomorphic homogeneous inside integral introduced invariant lemma mapping means meromorphic functions methods Moreover necessary notation obtain parameters particular periods poles polynomial positive possible presented problem Proof properties proved random walk rational relation Remark resp respect Riemann surface roots satisfies shown similar simple singular situation solution sufficient theorem transform unique unit circle universal covering values variables write yields zero