Moritz Gerlach

A new proof of Doobʼs theorem

Abstract We prove that every bounded, positive, irreducible, stochastically continuous semigroup on the space of bounded, measurable functions which is strong Feller, consists of kernel operators and possesses an invariant measure converges pointwise. This differs from Doobʼs theorem in that we do not require the semigroup to be Markovian and request a fairly weak kind of irreducibility. In addition, we elaborate on the various notions of kernel operators in this context, show the stronger result that the adjoint semigroup converges strongly and discuss as an example diffusion equations on rough domains. The proofs are based on the theory of positive semigroups and do not use probability theory.

Mean ergodic theorems on norming dual pairs

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.

On the peripheral point spectrum and the asymptotic behavior of irreducible semigroups of Harris operators

Given a positive, irreducible and bounded

On the lattice structure of kernel operators

C_0

Lower bounds and the asymptotic behaviour of positive operator semigroups

-semigroup on a Banach lattice with order continuous norm, we prove that the peripheral point spectrum of its generator is trivial whenever one of its operators dominates a non-trivial compact or kernel operator. For a discrete semigroup, i.e. for powers of a single operator

Semigroups of kernel operators

T