[Autour d'une approche analytique pour les marches à sauts arbitrairement grands dans le quart de plan]
Dans cette note, nous nous intéressons aux marches aléatoires avec sauts arbitrairement grands dans le quart de plan. Nous annonçons le développement, pour cette classe de modèles, de l'approche analytique proposée dans Fayolle et al. (1999) [4], initialement applicable aux marches à petits sauts dans le quart de plan. De nouvelles difficultés théoriques surgissent, qui, pour l'essentiel, sont abordées dans le cadre de la théorie des problèmes aux limites généralisés sur des surfaces de Riemann compactes.
In this note, we consider random walks in the quarter plane with arbitrary big jumps. We announce the extension to that class of models of the analytic approach of [4], initially valid for walks with small steps in the quarter plane. New technical challenges arise, most of them being tackled in the framework of generalized boundary value problems on compact Riemann surfaces.
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@article{CRMATH_2015__353_2_89_0,
author = {Fayolle, Guy and Raschel, Kilian},
title = {About a possible analytic approach for walks in the quarter plane with arbitrary big jumps},
journal = {Comptes Rendus. Math\'ematique},
pages = {89--94},
publisher = {Elsevier},
volume = {353},
number = {2},
year = {2015},
doi = {10.1016/j.crma.2014.11.015},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2014.11.015/}
}
TY - JOUR AU - Fayolle, Guy AU - Raschel, Kilian TI - About a possible analytic approach for walks in the quarter plane with arbitrary big jumps JO - Comptes Rendus. Mathématique PY - 2015 SP - 89 EP - 94 VL - 353 IS - 2 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2014.11.015/ DO - 10.1016/j.crma.2014.11.015 LA - en ID - CRMATH_2015__353_2_89_0 ER -
%0 Journal Article %A Fayolle, Guy %A Raschel, Kilian %T About a possible analytic approach for walks in the quarter plane with arbitrary big jumps %J Comptes Rendus. Mathématique %D 2015 %P 89-94 %V 353 %N 2 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2014.11.015/ %R 10.1016/j.crma.2014.11.015 %G en %F CRMATH_2015__353_2_89_0
Fayolle, Guy; Raschel, Kilian. About a possible analytic approach for walks in the quarter plane with arbitrary big jumps. Comptes Rendus. Mathématique, Tome 353 (2015) no. 2, pp. 89-94. doi : 10.1016/j.crma.2014.11.015. http://www.numdam.org/articles/10.1016/j.crma.2014.11.015/
[1] Walks with small steps in the quarter plane, Contemp. Math., Volume 520 (2010), pp. 1-39
[2] Boundary Value Problems in Queueing System Analysis, North-Holland Publishing Co., Amsterdam, 1983
[3] Two coupled processors: the reduction to a Riemann–Hilbert problem, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 47 (1979), pp. 325-351
[4] Random Walks in the Quarter Plane, Springer-Verlag, Berlin, 1999
[5] Analytic Combinatorics, Cambridge University Press, Cambridge, UK, 2009
[6] An analytical method in the theory of two-dimensional positive random walks, Sib. Math. J., Volume 13 (1972), pp. 1314-1329
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