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

Ben Goldys, Max Nendel, Michael Röckner

Published 2022-04-15Version 1

We define a class of not necessarily linear $C_0$-semigroups $(P_t)_{t\geq0}$ on $C_b(E)$ (more generally, on $C_\kappa(E):=\frac{1}{\kappa} C_b(E)$, for some continuous weight function $\kappa$) equipped with the mixed topology $\tau_1^\mathscr{M}$ for a large class of topological state spaces $E$. In the linear case we prove that such $(P_t)_{t\geq0}$ can be characterized as integral operators given by certain measure kernels. One main result is that 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 are jointly weakly continuous in space and time. 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 $(C_b(E), \tau_{1}^{\mathscr M})$. This solves a long standing open problem on Markov processes. Our results apply to a large number of Markov processes given as the laws of solutions to SDEs and SPDEs. We also introduce the notion of a Markov core operator $(L_0, D(L_0))$ for $(L, D(L))$ and prove that uniqueness of the Fokker-Planck-Kolmogorov equations given by $(L_0,D(L_0))$ for all Dirac initial conditions implies that $(L_0,D(L_0))$ is a Markov core operator for $(L,D(L))$. As a consequence we can identify the Kolmogorov operator of a large number of SDEs on finite and infinite dimensional state spaces as such. Furthermore, if each $P_t$ is merely convex, we prove that $(P_t)_{t \geq 0}$ gives rise to viscosity solutions to the Cauchy problem of its associated (non linear) infinitesimal generators. We also show that value functions of finite or infinite dimensional optimal control problems are examples of convex $C_0$-semigroups on $\left( C_b(E),\tau_{1}^{\mathscr M}\right)$.