Invariant Markov jump processes on compact groups

Abstract

The Markov jump processes on compact groups considered here are assumed to be invariant in the sense that the Markov transition probability function Pt,x can be defined by a convolution semi-group (πt)t⩾0 of probability measures on G as Pt,x = ϵx ∗πt, where ϵx is the Dirac measure concentrated on x. This semi-group is shown to be a {e}-Poisson semi-group and is determined by its generating functional φ, a measure on G such that - φ is a Poisson form. The processes are ergodic and the stationary probability distribution is related to the Haar measure on a closed subgroup H of G. Several properties of these processes, especially the spectral analysis of the Markov semi-group of bounded linear operators in C(G), are derived by the use of Fourier analysis. It is also shown that the jump processes can approximate the invariant Feller processes on compact groups. A number of examples is provided for the discussion of further features and possible classification schemes.