We study the long-term behavior of positive operator semigroups on spaces of bounded functions and on spaces of signed measures; such semigroups frequently appear in the study of parabolic partial differential equations and in stochastic analysis. One of our main results is a Tauberian type theorem which characterizes the convergence of a strongly Feller semigroup - uniformly on compact sets or in total variation norm - in terms of an ergodicity condition for the fixed spaces of the semigroup and its dual. In addition, we present a generalization of a classical convergence theorem of Doob to semigroups with arbitrarily large fixed space.
Throughout, we require only very weak assumptions on the involved semigroups; in particular, none of our convergence theorems requires any time regularity assumptions. This is possible due to recently developed methods in the asymptotic theory of semigroups on Banach lattices, which focus on algebraic rather than topological properties of the semigroup.
Our results enable us to efficiently analyse the long-term behavior of various parabolic equations and systems, even if the coefficients of the corresponding differential operator are unbounded.