Time inhomogeneous Generalized Mehler Semigroups

A time inhomogeneous generalized Mehler semigroup on a real separable Hilbert space $\text{mathds}H$ is defined through $$p_{s,t}f(x)={\int }_{\text{mathds}H}f(U(t,s)x+y){\mu }_{t,s}(dy),t\ge s,x\in \text{mathds}H$$ for every bounded measurable function $f$ on $\text{mathds}H$, where $(U(t,s){)}_{t\ge s}$ is an evolution family of bounded operators on $\text{mathds}H$ and $({\mu }_{t,s}{)}_{t\ge s}$ is a family of probability measures on $({\mathds{H}}, \B({\mathds{H}}))$satisfyingthetimeinhomogeneousskewconvolutionequations$${\mu }_{t,s}={\mu }_{t,r}\ast ({\mu }_{r,s}\circ U(t,r{)}^{-1}),t\ge r\ge s\mathrm{.}$$ This kind of semigroup is closely related with the transition semigroup" of non-autonomous (possibly non-continuous) Ornstein-Uhlenbeck process driven by some proper additive process. We show the weak continuity, infinite divisibility, associated “additive processes”, L'evy-Khintchine type representation, construction and spectral representation of $(\mu_{t,s})_{t\geq s}$. We study the structure, existence and uniqueness of the corresponding evolution systems of measures (=space-time invariant measures) of $(p_{s,t}{)}_{t\ge s}$. We also establish dimension free Harnack inequalities in the sense of Wang (1997, PTRF) for $(p_{s,t}{)}_{t\ge s}$. As applications of the Harnack inequalities, we investigate the strong Feller property and contractivity etc. for $p_{s,t}$. Finally we prove a Harnack inequality and show the strong Feller property for the transition semigroup of a semi-linear non-autonomous Ornstein-Uhlenbeck process driven by a Wiener process.