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 embeddingH⊂E 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.
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Bogachev, V.I., Röckner, M. & Schmuland, B. Generalized Mehler semigroups and applications. Probab. Th. Rel. Fields 105, 193–225 (1996). https://doi.org/10.1007/BF01203835
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DOI: https://doi.org/10.1007/BF01203835