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.
T. Kato [5] found an important property of semi-Fredholm pencils, now called the Kato decomposition. M. A. Kaashoek [3] introduced operators having the property P(S:k) as a generalization of semi-Fredholm operators. In this work, we study this class of operators. We show that it is characterized by a Kato-type decomposition. Other properties are also proved.
In analogy to the classical isomorphism between ((ℝⁿ), and (resp. and ), we show that a large class of moderate linear mappings acting between the space of compactly supported generalized functions and (ℝⁿ) of generalized functions (resp. the space of Colombeau rapidly decreasing generalized functions and the space of temperate ones) admits generalized integral representations, with kernels belonging to specific regular subspaces of (resp. ). The main novelty is to use accelerated...
We investigate the structure of the set of solutions of the Cauchy problem x’ = f(t,x), x(0) = x₀ in Banach spaces. If f satisfies a compactness condition expressed in terms of measures of weak noncompactness, and f is Pettis-integrable, then the set of pseudo-solutions of this problem is a continuum in , the space of all continuous functions from I to E endowed with the weak topology. Under some additional assumptions these solutions are, in fact, weak solutions or strong Carathéodory solutions,...
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.