Random motions, classes of ergodic Markov chains and beta distributions

We consider classes of discrete time Markov chains with continuous state space, the interval (0,1). These chains arise as stochastic models of phenomena in areas such as population theory, motion of particles in a random environment, etc. We exploit the Fréchet-Shohat theorem to establish that these Markov chains are ergodic and find explicitly their ergodic distributions as being beta distributions. Then we show that the convergence in total variation norm is at a geometric rate. Related topics are also discussed.