Random walks in ( + ) 2 with non-zero drift absorbed at the axes
Bulletin de la Société Mathématique de France, Volume 139 (2011) no. 3, p. 341-387

Spatially homogeneous random walks in ( + ) 2 with non-zero jump probabilities at distance at most 1, with non-zero drift in the interior of the quadrant and absorbed when reaching the axes are studied. Absorption probabilities generating functions are obtained and the asymptotic of absorption probabilities along the axes is made explicit. The asymptotic of the Green functions is computed along all different infinite paths of states, in particular along those approaching the axes.

Dans cet article, nous étudions les marches aléatoires du quart de plan ayant des sauts à distance au plus un, avec un drift non nul à l'intérieur et absorbées au bord. Nous obtenons de façon explicite les séries génératrices des probabilités d'absorption au bord, puis leur asymptotique lorsque le site d'absorption tend vers l'infini. Nous calculons également l'asymptotique des fonctions de Green le long de toutes les trajectoires, en particulier selon celles tangentes aux axes.

DOI : https://doi.org/10.24033/bsmf.2611
Classification:  60G50,  60G40,  30E20,  30F10
Keywords: random walk, Green functions, absorption probabilities, singularities of complex functions, holomorphic continuation, steepest descent method
@article{BSMF_2011__139_3_341_0,
     author = {Kurkova, Irina and Raschel, Kilian},
     title = {Random walks in $(\mathbb {Z}\_{+})^{2}$ with non-zero drift absorbed at the axes},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {139},
     number = {3},
     year = {2011},
     pages = {341-387},
     doi = {10.24033/bsmf.2611},
     zbl = {1243.60042},
     mrnumber = {2869310},
     language = {en},
     url = {http://www.numdam.org/item/BSMF_2011__139_3_341_0}
}
Random walks in $(\mathbb {Z}_{+})^{2}$ with non-zero drift absorbed at the axes. Bulletin de la Société Mathématique de France, Volume 139 (2011) no. 3, pp. 341-387. doi : 10.24033/bsmf.2611. http://www.numdam.org/item/BSMF_2011__139_3_341_0/

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