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Generalized Mehler semigroups and applications. (English) Zbl 0849.60066
Summary: We construct and study generalized Mehler semigroups \((p_t)_{t \geq 0}\) and their associated Markov processes \({\mathbf M}\). The construction methods for \((p_t)_{t \geq 0}\) 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 \(H \subset E\) 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 \((p_t)_{t \geq 0}\) with corresponding Markov process \({\mathbf M}\), the associated (non-symmetric) Dirichlet forms \(({\mathcal E}, D({\mathcal E}))\) are explicitly calculated and a necessary and sufficient condition for path regularity of \({\mathbf M}\) in terms of \(({\mathcal E}, D({\mathcal E}))\) is proved. Then, using Dirichlet form methods it is shown that \({\mathbf M}\) 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.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
47D07 Markov semigroups and applications to diffusion processes
31C25 Dirichlet forms
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G15 Gaussian processes
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