Mathematics – Probability
Scientific paper
2010-09-27
Mathematics
Probability
37 pages; correct some misprints
Scientific paper
A non-time-homogeneous generalized Mehler semigroup on a real separable Hilbert space \H is defined through [ p_{s,t}f(x)=\int_\H f(U(t,s)x+y) \mu_{t,s}(dy), \quad t\geq s, x\in\H,] for every bounded measurable function $f$ on $\H$, where $(U(t,s))_{t\geq s}$ is an evolution family of bounded operators on $\H$ and $\mu_{t,s}$ is a family of probability measures on $(\H, \B(\H))$ satisfying $\mu_{t,s}=\mu_{t,r}*(\mu_{r,s}\circ U(t,r)^{-1})$ for $t\geq r\geq s$. This kind of semigroups 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 infinite divisibility and a L\'evy-Khintchine type representation of $\mu_{t,s}$. We also study the corresponding evolution systems of measures (=space-time invariant measures), dimension free Harnack inequality and their applications to derive important properties of $p_{s,t}$. We also prove the Harnack inequality and show the strong Feller property for the transition semigroup of semi-linear non-autonomous Ornstein-Uhlenbeck processes driven by a Wiener process.
Ouyang Shun-Xiang
Röckner Michael
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