A new sufficient condition for the asymptotic stability of a locally Lipschitzian Markov semigroup acting on the space of signed measures is proved. This criterion is applied to the semigroup of Markov operators generated by a Poisson driven stochastic differential equation.
This work is concerned with the existence and regularity of solutions to the Neumann problem associated with a Ornstein–Uhlenbeck operator on a bounded and smooth convex set K of a Hilbert space H. This problem is related to the reflection problem associated with a stochastic differential equation in K.
Replacing the gaussian semigroup in the heat kernel estimates by the Ornstein-Uhlenbeck semigroup on , we define the notion of Kolmogorov kernel estimates. This allows us to show that under Dirichlet boundary conditions Ornstein-Uhlenbeck operators are generators of consistent, positive, (quasi-) contractive -semigroups on for all and for every domain . For exterior domains with sufficiently smooth boundary a result on the location of the spectrum of these operators is also given.