Invariant densities and mean ergodicity of Markov operators (Q1424118)

A semigroup of operators on a space \(L^1(\Omega,\Sigma,\mu)\) is Markov if the set of probability densities in \(\Omega\) is invariant for all operators in the semigroup. For a strongly continuous Markov semigroup of operators \(\mathcal T:=\{T_t\}_{t\geq0}\), the following conditions are equivalent: (i) \(\mathcal T\) has an invariant density; (ii) \(\limsup_{t\to\infty} \| f-T_tf\| <2\) for some density \(f\); (iii) \(\limsup_{t\to\infty} \| d-T_tg\| <2\) for some pair of densities \(d\), \(g\). \(\mathcal T\) is mean ergodic if \(\lim_{t\to\infty} [\frac{1}{\tau}\int_0^\tau T_t\,dt] f\) exists in \(L^1\) for all \(f\). It is weakly almost periodic if for each \(f\) the orbit \(\{T_tf\}_t\) is conditionally weakly compact (i.e. any sequence has a weakly Cauchy subsequence). It is shown that any mean ergodic bounded semigroup of positive operators \(\mathcal T\) on \(L^1(\Omega)\) is weakly almost periodic. The Frobenius-Perron operator \(P\) corresponding to a nonsingular (i.e. absolutely continuous) transformation \(\tau:\Omega\rightarrow\Omega\) is defined through the Radon-Nikodým theorem by \(\int_A Pf\,d\mu= \int_{\tau^{-1}(A)}f\,d\mu\). Any Frobenius-Perron operator is Markov, and when \(\{\tau_t\}_t\) is a family of nonsingular transformations, \(\{P_{\tau_t}\}_t\) is the Frobenius-Perron semigroup. For such a semigroup the conditions of weakly almost periodicity or mean ergodicity are equivalent to the existence of a density \(w\) such that \(\limsup_{t\to\infty} \| P_{\tau_t}f-w\|<2 \) for every density \(f\).