Time inhomogeneous generalized Mehler semigroups and skew convolution equations
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Shun-Xiang Ouyang
und
Michael Röckner
Abstract
A time inhomogeneous generalized Mehler semigroup on a real separable Hilbert space ℍ is defined through ps,tf(x) = ∫ℍf(U(t,s)x + y)μt,s(dy), 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 (μt,s)t≥s is a family of probability measures on (ℍ,ℬ(ℍ)) satisfying the following time inhomogeneous skew convolution equations: μt,s = μt,r * (μr,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 μt,s converges weakly to δ0 as t ↓ s or s ↑ t. We show that μt,s has further weak continuity properties in t and s. As a consequence, we prove that for every t ≥ s, μt,s is infinitely divisible. Natural stochastic processes associated with (μt,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 (ps,t)t≥s. We also establish a dimension free Harnack inequality for (ps,t)t≥s and present some of its applications.
Funding source: DFG
Award Identifier / Grant number: SFB-701
Funding source: DFG
Award Identifier / Grant number: IRTG-1132
The authors thank Alexander Grigor'yan and Zenghu Li for useful discussions. Furthermore, we thank the referee for his careful reading of the paper and valuable suggestions for improving this paper.
© 2016 by De Gruyter
Artikel in diesem Heft
- Frontmatter
- Ring class invariants over imaginary quadratic fields
- The structure of p-local compact groups of rank 1
- Sampling and interpolation on some nilpotent Lie groups
- Topology of moduli spaces of free group representations in real reductive groups
- Finite symmetries of S4
- Variations of Minkowski's theorem on successive minima
- A one-relator group with long lower central series
- On transfinite nilpotence of the Vogel–Levine localization
- Time inhomogeneous generalized Mehler semigroups and skew convolution equations
- Specifying the Auslander–Reiten translation for complexes of modules
- Second order elliptic operators with diffusion coefficients growing as |x|α at infinity
- On the fourth derivative test for exponential sums
Artikel in diesem Heft
- Frontmatter
- Ring class invariants over imaginary quadratic fields
- The structure of p-local compact groups of rank 1
- Sampling and interpolation on some nilpotent Lie groups
- Topology of moduli spaces of free group representations in real reductive groups
- Finite symmetries of S4
- Variations of Minkowski's theorem on successive minima
- A one-relator group with long lower central series
- On transfinite nilpotence of the Vogel–Levine localization
- Time inhomogeneous generalized Mehler semigroups and skew convolution equations
- Specifying the Auslander–Reiten translation for complexes of modules
- Second order elliptic operators with diffusion coefficients growing as |x|α at infinity
- On the fourth derivative test for exponential sums
