Time inhomogeneous generalized Mehler semigroups and skew convolution equations

Abstract

Abstract A time inhomogeneous generalized Mehler semigroup on a real separable Hilbert space ℍ is defined through p s,t f(x) = ∫ℍ f(U(t,s)x + y)μ t,s (d y), s,t ∈ ℝ, t ≥ s, x ∈ ℍ, for every bounded measurable function f on ℍ, where (U(t,s)) t≥s is an evolution family of bounded operators on ℍ and (μ t,s ) t≥s is a family of probability measures on (ℍ,ℬ(ℍ)) satisfying the following time inhomogeneous skew convolution equations: μ t,s = μ t,r * (μ r,s ∘ 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 μ t,s converges weakly to δ0 as t ↓ s or s ↑ t. We show that μ t,s has further weak continuity properties in t and s. As a consequence, we prove that for every t ≥ s, μ t,s is infinitely divisible. Natural stochastic processes associated with (μ 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,t ) t≥s and present some of its applications.