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Random Walks in the Quarter Plane Absorbed at the Boundary : Exact and Asymptotic

# Random Walks in the Quarter Plane Absorbed at the Boundary : Exact and Asymptotic,Kilian Raschel

Random Walks in the Quarter Plane Absorbed at the Boundary : Exact and Asymptotic
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.
Published in 2009.
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## Citations (4)

### Martin boundary of a killed random walk on $\Z_+^d$(Citations: 1)

Published in 2009.

### The t-Martin boundary of reflected random walks on a half-space

Published in 2009.

### Green functions and Martin compactification for killed random walks related to SU(3)

Published in 2009.

### Passage time from four to two blocks in the voter model

Published in 2009.