Generalized Mehler semigroups and applications

Liouville theorems for non-local operators

“…Remark that Ornstein-Uhlenbeck operators are investigated also in infinite dimensions, see for instance [10,15,20,28,41,47,48,58]. A challenging task would be to apply other probabilistic techniques, based on Malliavin calculus (see [8,37,43]) or on coupling (see [14,55]) to non-local operators.…”

Continuity and generators of dynamical semigroups for infinite Bose systems

“…It is worth remarking that convolution semigroups satisfying a relation similar to (11) were studied independently in the context of generalized Mehler semigroups, see [8,18,34]. A main result of [7] was the first part of the following theorem which we generalize here by allowing the symplectic form σ to be degenerate.…”

Hypercontractivity and Absolute Continuity of Gaussian Generalized Mehler Semigroups in Hilbert Spaces

“…The Gaussian SC-semigroup constructed in Example 4.1 is differentiable, i.e. the function t → µ t (a) is differentiable at t = 0 (see [1]). However, a rich class of non-differentiable Gaussian SC-semigroups arise in the study of catalytic branching superprocesses with immigration (see [5]).…”

“…A characterization for Gaussian Mehler semigroups was given by Bogachev et al [1] under the differentiability condition of the function t → µ t (a) at t = 0. A necessary and sufficient condition for a Gaussian SC-semigroup to be differentiable was given in van Neerven [11] .…”

“…Generalized Mehler semigroups were introduced by Bogachev et al [1] in studying nonsymmetric Ornstein-Uhlenbeck processes. Those processes are closely related to the immigration processes formulated by SC-semigroups in the measure-valued setting (see [4,5,8,9]).…”