Generalized Mehler semigroups and applications
We construct and study generalized Mehler semigroups (p(t))(t greater than or equal to 0) and their associated Markov processes M. The construction methods for (p(t))(t greater than or equal to 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 subset of E being Hilbert-Schmidt. The same analytic extension results are applied to construct strong solutions to stochastic differential equations of type dX(t) = CdWt + 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 greater than or equal to 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. Thee, 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.
105
2
193-225
193-225
Springer