Operator semigroups in the mixed topology and the infinitesimal description of Markov processes

Abstract

We define a class of not necessarily linear $C_0$-semigroups $(P_t)_{t\geq0}$ on $C_b(E)$ (more generally, on $C_\kappa(E):=\frac1\kappa C_b(E)$, for some bounded function $\kappa$, which is the pointwise limit of a decreasing sequence of continuous functions) equipped with the mixed topology $\tau_1^{\mathscr M}$ for a large class of topological 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 $(C_b(E),\tau_1^{\mathscr M})$. In the linear case, we characterize such $(P_t)_{t\geq0}$ as integral operators given by measure kernels satisfying certain tightness properties. As a consequence, transition semigroups of Markov processes are $C_0$-semigroups on $(C_b(E),\tau_1^{\mathscr M})$, if they leave $C_b(E)$ 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. Furthermore, we introduce the notion of a Markov core operator $(L_0,D(L_0))$ for the above generators $(L,D(L))$ and prove that uniqueness of the Fokker-Planck-Kolmogorov equations corresponding to $(L_0,D(L_0))$ for all Dirac initial conditions implies that $(L_0,D(L_0))$ is a Markov core operator for $(L,D(L))$. If each $P_t$ is merely convex, we prove that $(P_t)_{t\geq0}$ 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 $(C_\kappa(E),\tau_\kappa^{\mathscr M})$.