Generalized Mehler semigroups and applications

Abstract

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.

On analytic Ornstein-Uhlenbeck semigroups in infinite dimensions

“…The next theorem generalizes results which were proved by Fuhrman [5], and Bogachev, Röckner and Schmuland [1] in a Hilbert space setting.…”

Uniqueness for solutions of Fokker–Planck equations related to singular SPDE driven by Lévy and cylindrical Wiener noise

“…It belongs to the family of so-called generalized Mehler semigroups, which have been introduced in [BRS96] and studied e.g. in [FR00], [SS01], [LR02], [App07] (see also the references therein).…”

Branching Processes with Immigration and Related Topics

“…Example 2.3 Let us consider the case where S = H is a real separable Hilbert space and Q t (x, ·) ≡ δ Ttx for a strongly continuous semigroup of bounded linear operators (T t ) t≥0 on H. In this case, (Q γ t ) t≥0 is called a generalized Mehler semigroup associated with (T t ) t≥0 , which corresponds to a generalized Ornstein-Uhlenbeck process (OU-process). This formulation of the processes was given by Bogachev et al [4]; see also [16,25].…”

“…Based on Theorem 7.1 they showed that each centered SC-semigroup is uniquely determined by an infinitely divisible probability measure on the entrance spaceH for the semigroup (T t ) t≥0 , which is an enlargement of H. They proved that a centered SC-semigroup can always be extended to a regular one on the entrance space. Those results provide an approach to the study of irregular generalized Mehler semigroups with which one can reduce some of their analysis to the framework of [4,25,67].…”