PUB - Publikationen an der Universität Bielefeld

We define a class of not necessarily linear C 0-semigroups (P t ) t >= 0 on Cb(E) b (E) (more generally, on C kappa kappa (E) := kappa 1 Cb(E), b (E) , for some bounded function kappa , which is the pointwise limit of a decreasing sequence of continuous functions) equipped with the mixed topology tau 1 M for a large class of topological (not necessarily Polish) state spaces E . If these semigroups are linear, classical theory of operator semigroups on locally convex spaces as well as the theory of bicontinuous semigroups apply to them. In particular, they are infinitesimally generated by their generator (L, D(L)) and thus reconstructable through an Euler formula from their strong derivative at zero in (Cb(E), b (E), tau 1 M ) . In the linear case, we prove that such (P t ) t >= 0 can be characterized as integral operators given by measure kernels satisfying certain tightness properties. As a consequence, transition semigroups of Markov processes are C 0-semigroups on (Cb(E), b (E), tau 1 M ) , if they leave Cb(E) b (E) invariant and they are jointly weakly continuous in space and time. Hence, they can be reconstructed from their strong derivative at zero and thus have a fully infinitesimal description. This solves an open problem for Markov processes. We show that our results apply to a large number of Markov processes, e.g., those given as the laws of solutions to SDEs and SPDEs, including the stochastic 2D Navier-Stokes equations and the stochastic fast and slow diffusion porous media equations. Furthermore, we introduce the notion of a Markov core operator (L0, 0 , D(L0)) 0 )) for the above generators (L, D(L)) and prove that uniqueness of the Fokker-Planck-Kolmogorov equations corresponding to (L0, 0 , D(L0)) 0 )) for all Dirac initial conditions implies that (L0, 0 , D(L0)) 0 )) is a Markov core operator for (L, D(L)). . As a consequence, we identify the Kolmogorov operators of a large number of SDEs on finite and infinite dimensional state spaces as Markov core operators for the infinitesimal generators of the C 0-semigroups on (C kappa kappa (E), & ccedil;') given by their transition semigroups. Furthermore, if each Pt t is merely convex, we prove that (P t ) t >= 0 gives rise to viscosity solutions to the Cauchy problem given by its associated (nonlinear) infinitesimal generator. We also show that value functions of optimal control problems, both, in finite and infinite dimensions are particular instances of convex C 0-semigroups on (C kappa kappa (E), t kappa'). kappa ') . (c) 2024 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).