On the hypergroup property

Abstract

The hypergroup property satisfied by certain reversible Markov chains can be seen as a generalization of the convolution related features of class random walks on groups. Carlen, Geronimo and Loss~\cite{MR2764893} developed a method for checking this property in the context of Jacobi eigen-polynomials. A probabilistic extension of their approach is proposed here, enabling to recover the discrete example of the biased Ehrenfest model due to Eagleson \cite{MR0328162}. Next a spectral characterization is provided for finite birth and death chains enjoying the hypergroup property with respect to one of the boundary points.