2010/09/27 by Shun-Xiang Ouyang, Michael Röckner, Ouyang, Shun-Xiang +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #Geometric Analysis and Curvature Flows #Spectral Theory in Mathematical Physics #Stochastic processes and financial applications #math.PR #msc:47D07 #msc:60J75
paper · pdf · doi:10.48550/arxiv.1009.5314
93 pages; corrected and extended
arxiv created 2012/09/10 · arxiv updated 2012/09/12
A time inhomogeneous generalized Mehler semigroup on a real separable Hilbert space \mathdsH is defined through ps,tf(x)=∫_\mathdsH f(U(t,s)x+y) μt,s(dy), t≥ s, x∈\mathdsH for every bounded measurable function f on \mathdsH, where (U(t,s))t≥ s is an evolution family of bounded operators on \mathdsH and (μt,s)t≥ s is a family of probability measures on (\mathdsH, \B(\mathdsH)) satisfying the time inhomogeneous skew convolution equations μt,s=μt,r*(μr,s∘ U(t,r)-1), t≥ r≥ 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", Lévy-Khintchine type representation, construction and spectral representation of (μt,s)t≥ s. We study the structure, existence and uniqueness of the corresponding evolution systems of measures (=space-time invariant measures) of (ps,t)t≥ s. We also establish dimension free Harnack inequalities in the sense of Wang (1997, PTRF) for (ps,t)t≥ s. As applications of the Harnack inequalities, we investigate the strong Feller property and contractivity etc. for ps,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.