Generalized Mehler semigroups and applications

We construct and study generalized Mehler semigroups (pt)t≥0 and their associated Markov processes M. The construction methods for (pt)t≥0 are based on some new purely functional analytic results implying, in particular, that any strongly continuous semigroup on a Hilbert space H can be extended to some larger Hilbert space E, with the embedding H ⊂ E being Hilbert-Schmidt. The same analytic extension results are applied to construct strong solutions to stochastic differential equations of type dXt = CdWt + AXtdt (with possibly unbounded linear operators A and C on H) on a suitably chosen larger space E. For Gaussian generalized Mehler semigroups (pt)t≥0 with corresponding Markov process M, the associated (non-symmetric) Dirichlet forms (E , D(E)) are explicitly calculated and a necessary and sufficient condition for path regularity of M in terms of (E , D(E)) is proved. Then, using Dirichlet form methods it is shown that M weakly solves the above stochastic differential equation if the state space E is chosen appropriately. Finally, we discuss the differences between these two methods yielding strong resp. weak solutions. AMS Subject Classification (1991) Primary: 47D07, 60H15 Secondary: 31C25, 60H10, 60G15 1 Department of Mechanics and Mathematics, Moscow State University, 119899 Moscow, Russia. 2 Department of Mathematics, University of Bielefeld, D–33615 Bielefeld, Germany. 3 Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1.