Stochastic processes with orthogonal polynomial eigenfunctions (Q1035624)

The author studies time-homogeneous reversible stationary Markov processes such that the eigenfunctions of the associated transition operators form a system of orthogonal polynomials with the stationary distribution as orthogonality measure. There exist well-known examples with Gamma, Normal, Poisson, Negative Binomial, and Beta distributions as stationary distributions. In the paper under review, the author presents representations of such stationary processes with a given stationary distribution of the above type in terms of subordinations of basic building blocks and using usual Levy processes on \([0,\infty[\) as subordinators. In particular for the Beta case with Jacobi polynomials as eigenfunctions, this representation is related to the Levy-Khintchine formula on associated convolution structures.