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

Beniamin Goldys, Max Nendel and Michael Röckner

Abstract

We define a class of not necessarily linear $C_0$-semigroups $(P_t{)}_{t\ge 0}$ on $C_b(E)$ (more generally, on $C_{\kappa }(E):=\frac{1}{\kappa }{C}_b(E)$, for some growth bounding continuous function $\kappa$) equipped with the mixed topology ${\tau }_1^{\mathcal{M}}$ for a large class of topological state spaces $E$. In the linear case we prove that such $(P_t{)}_{t\ge 0}$ can be characterized as integral operators given by measure kernels satisfying certain properties. We prove that the strong and weak infinitesimal generators of such $C_0$-semigroups coincide. As a main result we prove that transition semigroups of Markov processes are $C_0$-semigroups on $(C_b(E),{\tau }_1^{\mathcal{M}})$, if they leave $C_b(E)$ invariant and they 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 $(C_b(E),{\tau }_1^{\mathcal{M}})$. 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, 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 $(L_0,D(L_0))$ 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 $(L,D(L))$. As a consequence we can identify the Kolmogorov operator 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 $(C_{\kappa }(E),{\tau }_{\kappa }^{\mathcal{M}})$ given by their transition semigroups. Furthermore, if each $P_t$ is merely convex, we prove that $(P_t{)}_{t\ge 0}$ gives rise to viscosity solutions to the Cauchy problem of its associated (non linear) infinitesimal generators. 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 $(C_{\kappa }(E),{\tau }_{\kappa }^{\mathcal{M}})$.

Keywords: Markov process, stochastic (partial) differential equation, mixed topology, strongly continuous semigroup, infinitesimal generator, Markov uniqueness, viscosity solution, Fokker-Planck-Kolmogorov equation, generalized Mehler semigroups, Levy-Khintchin representation.

AMS Subject Classification: Primary 47D06;; secondary 47H20; 60H10; 60H15; 60J25; 60J35.