Convex monotone semigroups on spaces of continuous functions

Abstract

Motivated by model uncertainty and stochastic control problems, this thesis aims to develop a systematic theory for strongly continuous convex monotone semigroups on spaces of continuous functions. The present approach is self-contained and does, in particular, not rely on the theory of viscosity solutions. Instead, we provide a comparison principle for semigroups related to Hamilton-Jacobi-Bellman equations which uniquely determines the semigroup by its infinitesimal generator evaluated at smooth functions. While the statement itself resembles the classical analogue for strongly continuous linear semigroups, the proof requires the introduction of several novel analytical concepts such as the Lipschitz set and the $\Gamma$-generator. Furthermore, we provide general approximation schemes and stability results for nonlinear semigroups. While the focus of this thesis is on the development of a systematic theory for convex monotone semigroups, the abstract results are all motivated by applications and therefore providing conditions that can easily be verified is also a major contribution of this work.