  A time inhomogeneous generalized Mehler semigroup on a real separable Hilbert
space ${\mathds{H}}$ is defined through $$ p_{s,t}f(x)=\int_{\mathds{H}}
f(U(t,s)x+y)\,\mu_{t,s}(dy), \quad t\geq s, \ x\in{\mathds{H}} $$ 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
$(\mu_{t,s})_{t\geq s}$ is a family of probability measures on $({\mathds{H}},
\B({\mathds{H}}))$ satisfying the time inhomogeneous skew convolution equations
$$\mu_{t,s}=\mu_{t,r}*(\mu_{r,s}\circ U(t,r)^{-1}),\quad t\geq r\geq s.$$ 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", 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
$(p_{s,t})_{t\geq s}$. 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.
