On a longstanding open problem in the theory of Markov processes
A talk in the Bielefeld Stochastic Afternoon series by
Michael Röckner
| Abstract: | We define a class of not necessarily linear $C_0$-semigroups $(P_t)_{t\geq0}$ on $C_b(E)$ (more generally, on $C_\kappa(E):=\frac{1}{\kappa} C_b(E)$,
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 $(P_t)_{t\geq0}$ 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.
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.
Furthermore, we prove that each $P_t$ has a stochastic representation as a convex expectation in terms of a nonlinear Markov process.
Within the CRC this talk is associated to the project(s): A5, B1 |