Time inhomogeneous Generalized Mehler Semigroups

A time inhomogeneous generalized Mehler semigroup on a real separable Hilbert space ${\mathds{H}}$ is defined through $$ p_{s,t}f(x)=\int_{\mathds{H}} f(U(t,s)x+y)\,μ_{t,s}(dy), \quad t\geq s, \ x\in{\mathds{H}} $$ for every bounded measurable function $f$ on ${\mathds{H}}$, where $(U(t,s))_{t\geq s}$ is an evolution family of bounded operators on ${\mathds{H}}$ and $(μ_{t,s})_{t\geq s}$ is a family of probability measures on $({\mathds{H}}, \B({\mathds{H}}))$ satisfying the time inhomogeneous skew convolution equations $μ_{t,s}=μ_{t,r}*(μ_{r,s}\circ U(t,r)^{-1}),\quad t\geq r\geq s.$ 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", Lévy-Khintchine type representation, construction and spectral representation of $(μ_{t,s})_{t\geq s}$. We study the structure, existence and uniqueness of the corresponding evolution systems of measures (=space-time invariant measures) of $(p_{s,t})_{t\geq s}$. We also establish dimension free Harnack inequalities in the sense of Wang (1997, PTRF) for $(p_{s,t})_{t\geq s}$. 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.

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