Random walks in the quarter plane absorbed at the boundary: exact and asymptotic

Abstract : Nearest neighbor random walks in the quarter plane that are absorbed when reaching the boundary are studied. The cases of positive and zero drift are considered. Absorption probabilities at a given time and at a given site are made explicit. The following asymptotics for these random walks starting from a given point $(n_0,m_0)$ are computed : that of probabilities of being absorbed at a given site $(i,0)$ [resp. $(0,j)$] as $i\to \infty$ [resp. $j \to \infty$], that of the distribution's tail of absorption time at $x$-axis [resp. $y$-axis], that of the Green functions at site $(i,j)$ when $i,j\to \infty$ and $j/i \to \tan \gamma$ for $\gamma \in [0, \pi/2]$. These results give the Martin boundary of the process and in particular the suitable Doob $h$-transform in order to condition the process never to reach the boundary. They also show that this $h$-transformed process is equal in distribution to the limit as $n\to \infty$ of the process conditioned by not being absorbed at time $n$. The main tool used here is complex analysis.
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Preprints, Working Papers, ...
2009
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https://hal.archives-ouvertes.fr/hal-00361951
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Submitted on : Monday, February 16, 2009 - 10:48:44 PM
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Kilian Raschel. Random walks in the quarter plane absorbed at the boundary: exact and asymptotic. 2009. <hal-00361951>

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