A new proof of Doob's theorem

TL;DR

This paper presents a new proof of Doob's theorem for a class of semigroups on bounded measurable functions, relaxing some conditions and using positive semigroup theory instead of probability theory.

Contribution

It provides a novel proof of Doob's theorem that does not require the semigroup to be Markovian and introduces a weaker form of irreducibility.

Findings

Semigroup convergence pointwise established

Strong convergence of the adjoint semigroup shown

Application to diffusion equations on rough domains

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 Doob's 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.