Mean ergodic theorems on norming dual pairs

Abstract

We extend the classical mean ergodic theorem to the setting of norming dual pairs. It turns out that, in general, not all equivalences from the Banach space setting remain valid in our situation. However, for Markovian semigroups on the norming dual pair (C_b(E), M(E)) all classical equivalences hold true under an additional assumption which is slightly weaker than the e-property.Comment: 18 pages, 1 figur

On the lattice structure of kernel operators

“…Our strategy is similar to the one in [11], however due to the abstract results of Sections 2 and 3 the proof simplifies as we can work directly within the lattice of kernel operators. This complements recent results about mean ergodicity of semigroups of kernel operators, see [10] and [9].…”

A Tauberian theorem for strong Feller semigroups

“…Let us assume (i) and pick µ ∈ M (Ω). It follows from [4,Thm 5.7] that there existsμ ∈ fix(T ) such that lim A t µ −μ, f = 0 for all f ∈ C b (Ω), i.e. assertion (ii) holds.…”

“…convergence of the Cesàro averages in the weak topology induced by the bounded continuous functions. In [4] M. Kunze and the author characterized ergodicity of semigroups on general norming dual pairs in the spirit of the classical mean ergodic theorem. In particular for eventually strong Feller Markov semigroups it was shown that they are weakly ergodic if the space of invariant measures separates the space of invariant continuous functions.…”

“…Making use of the characterization of weak ergodicity in [4,Thm 5.7], we obtain the following Corollary. …”

Diffusion with nonlocal Dirichlet boundary conditions on domains

“…Moreover, in view of Proposition 2.8, we have fix(T µ ) = span{1}. We now have to distinguish the situation where the semigroup T µ is weakly ergodic (in the sense of [19]) and the situation where it is not weakly erdodic. As T µ enjoys the strong Feller property, we infer from [19,Theorems 4.4 and 5.7] that T µ is weakly ergodic if and only if fix(T µ ) ′ separates fix(T µ ).…”