Lipschitz contractions, unique ergodicity and asymptotics of Markov semigroups (Q2910040)

Let \(K\) be a compact metric space. A favorite approach to study asymptotics and further qualitative properties of operators (and, in particular, operator semigroups) on \(C(X)\) is to introduce \(L^2\)-realizations of their generators, show that they generate Markov semigroups on \(L^2\) and check corresponding properties by suitable energy methods and, finally, extrapolate these properties to the original functional setting.NEWLINENEWLINEIn the present paper, the setting is somewhat dual. One namely considers a Markov operator in \(C(X)\) and tries to show that some relevant properties are also enjoyed by its realizations in all \(L^p\)-spaces. More precisely, one considers simultaneously the cases of a semigroup \((T(t))_{t\geq 0}\) of Markov operators on \(C(X)\) along with the powers \((T^n)_{n\in \mathbb N}\) of a bounded linear Markov operator. Then, the authors show in their main result in Section 2 that, if (i) each operator in either net leaves invariant the Lipschitz continuous functions and (ii) their Lipschitz seminorms converge to \(0\) as the time \(t\) or \(n\) goes to \(+\infty\), then the net converges in the \(\sup\)-norm towards a projection of rank one -- the averaging operator with respect to a unique (probability) invariant measure -- and so does the net of the \(L^p\)-realizations of the same operators, with respect to the same probability measure. An estimate for the convergence rate is also given. Interestingly, these results do not depend on irreducibility of the involved operators.NEWLINENEWLINEWhile condition (ii) on convergence of the Lipschitz seminorm is often easily checked if one considers the sequence of iterates of a Markov operator, this is not necessarily the case if one works instead with semigroups. The authors propose, in particular, a sufficient condition that ensures validity of (ii), based on approximability by iterates of further Markov operators. This condition is tailored for the investigation of Kantorovitch operators -- these have been discussed by the same authors in several previous papers, and indeed this example is treated in detail in Section 3.1. Another application to a class of generators of (generalized) Fleming-Viot processes is displayed in Section 3.2.