Lipschitz Contractions, Unique Ergodicity and Asymptotics of Markov Semigroups

Altomare, Francesco ; Raşa, Ioan

Bollettino della Unione matematica italiana, Série 9, Tome 5 (2012) no. 1, pp. 1-17

Résumé

We are mainly concerned with the asymptotic behaviour of both discrete and continuous semigroups of Markov operators acting on the space $C(X)$ of all continuous functions on a compact metric space $X$. We establish a simple criterion under which such semigroups admit a unique invariant probability measure $\mu$ on $X$ that determines their limit behaviour on $C(X)$ and on $L^{p}(X; \mu)$. The criterion involves the behaviour of the semigroups on Lipschitz continuous functions and on the relevant Lipschitz seminorms. Finally, we discuss some applications concerning the Kantorovich operators on the hypercube and the Bernstein-Durrmeyer operators with Jacobi weights on $[0; 1]$. As a consequence we determine the limit of the iterates of these operators as well as of their corresponding Markov semigroups whose generators fall in the class of Fleming-Viot differential operators arising in population genetics.