Abstract
A random walk on ℕd endowed with a structure of polynomial hypergroup, is a Markov chain which transition kernel takes the form of an operator \(T_\mu = \sum {_{n \in \mathbb{N}^d } \mu (n){\text{ }}T_n },\), where μ is a probability measure on ℕd and \((T_n )_{n \in \mathbb{N}^d } \) are the hypergroup translations. In this paper, we study the positive spectrum of the operator T μ , and we give an integral representation formula for its positive eigenvectors. Our result also contains the description of the Martin boundary. Then, we consider the particular case of the random walk to the nearest neighbour with reflecting barriers on the coordinate hyperplanes.
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Godefroy, L. Martin Boundary of Polynomial Random Walks on the d-Dimensional Lattice of Nonnegative Integers. Journal of Theoretical Probability 17, 111–129 (2004). https://doi.org/10.1023/B:JOTP.0000020477.86897.82
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DOI: https://doi.org/10.1023/B:JOTP.0000020477.86897.82