PUB - Publikationen an der Universität Bielefeld

A time inhomogeneous generalized Mehler semigroup on a real separable Hilbert space is defined through p(s,1)f(x) = integral(IH) f(U(t,s)x + y) mu(t,s)(dy), s,t epsilon IR, t >= s, x epsilon IH, for every bounded measurable function on IH, where (U(t, s))(t >= s) is an evolution family of bounded operators on IH and (mu(t,s) )(t >= s) is a family of probability measures on (IH, B(IH)) satisfying the following time inhomogeneous skew convolution equations: mu(t,s) = mu(t,r) * (mu(r,s) circle U(t,r)(-1), t >= r >= s. This kind of semigroups typically arise as the "transition semigroups" of non-autonomous (possibly non-continuous) Ornstein-Uhlenbeck processes driven by some proper additive process. Suppose that mu(t,s) converges weakly to delta(0) as t down arrow s or s up arrow t. We show that mu(t,s) has further weak continuity properties in t and s. As a consequence, we prove that for every t >= s, mu(t,s) is infinitely divisible. Natural stochastic processes associated with (mu(t,s))(t >= s) are constructed and are applied to get probabilistic proofs for the weak continuity and infinite divisibility. Then we analyze the structure, existence and uniqueness of the corresponding evolution systems of measures (= space-time invariant measures) of (p(s,t))(t >= s) .We also establish a dimension free Harnack inequality for (P-s,P-t )(t >= s) and present some of its applications.