Abstract. We construct and study generalized Mehler semigroups (p t ) t≥0 and their associated Markov processes M. 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 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 dX t = CdW t + AX t dt (with possibly unbounded linear operators A and C on H) on a suitably chosen larger space E. For Gaussian generalized Mehler semigroups (p t ) 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.
Abstract. 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 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 nondifferentiable skew convolution semigroups. Path regularity of the corresponding generalized Ornstein-Uhlenbeck processes in different topologies is also discussed. (2000): Primary 60J35, 60G20; Secondary 60G57, 60J80
Mathematics Subject Classifications
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