Positivity Preserving Forms have the Fatou Property

We prove that if (,D) is a positivity preserving form on L ² (E;m), and if (u _n)_n is a sequence in D() converging m-almost everywhere to u L ² (E;m), then (u,u) lim inf_n (u _n ,u _n ).     

Generalized Mehler Semigroups and Catalytic Branching Processes with Immigration

Skew convolution semigroups play an important role in the study of generalized Mehler semigroups and Ornstein–Uhlenbeck processes. We give a characterization for a general skew convolution semigroup on a real separable Hilbert space whose characteristic functional is not necessarily differentiable at the initial time. A connection between this subject and catalytic branching superprocesses is established through fluctuation limits, providing a rich class of non-differentiable skew convolution semigroups. Path regularity of the corresponding generalized Ornstein–Uhlenbeck processes in different topologies is also discussed.     

Generalized Mehler semigroups and applications

Summary We construct and study generalized Mehler semigroups (p _t ) _t 0 and their associated Markov processesM. The construction methods for (p _t ) _t 0 are based on some new purely functional analytic results implying, in particular, that any strongly continuous semigroup on a Hilbert spaceH can be extended to some larger Hilbert spaceE, with the embeddingHE being Hilbert-Schmidt. The same analytic extension results are applied to construct strong solutions to stochastic differential equations of typedX _t =C dW _t +AX _t dt (with possibly unbounded linear operatorsA andC onH) on a suitably chosen larger spaceE. For Gaussian generalized Mehler semigroups (p _t ) _t 0 with corresponding Markov processM, the associated (non-symmetric) Dirichlet forms (E D(E)) are explicitly calculated and a necessary and sufficient condition for path regularity ofM in terms of (E,D(E)) is proved. Then, using Dirichlet form methods it is shown thatM weakly solves the above stochastic differential equation if the state spaceE is chosen appropriately. Finally, we discuss the differences between these two methods yielding strong resp. weak solutions.