Dissipativity of the delay semigroup

Under mild conditions a delay semigroup can be transformed into a (generalized) contraction semigroup by modifying the inner product on the (Hilbert) state space into an equivalent inner product. Applications to stability of differential equations with delay and stochastic differential equations with delay are given as examples

Delay differential equations driven by Levy processes: stationarity and Feller properties

We consider a stochastic delay differential equation driven by a general Levy process. Both, the drift and the noise term may depend on the past, but only the drift term is assumed to be linear. We show that the segment process is eventually Feller, but in general not eventually strong Feller on the Skorokhod space. The existence of an invariant measure is shown by proving tightness of the segments using semimartingale characteristics and the Krylov-Bogoliubov method. A counterexample shows that the stationary solution in completely general situations may not be unique, but in more specific cases uniqueness is established.Comment: 28 page

Invariant measures for monotone SPDE's with multiplicative noise term

We study diffusion processes corresponding to infinite dimensional semilinear stochastic differential equations with local Lipschitz drift term and an arbitrary Lipschitz diffusion coefficient. We prove tightness and the Feller property of the solution to show existence of an invariant measure. As an application we discuss stochastic reaction diffusion equations.Comment: 10 page

Lexicographic cones and the ordered projective tensor product

We introduce lexicographic cones, a method of assigning an ordered vector space \Lex(S) to a poset $S$, generalising the standard lexicographic cone. These lexicographic cones are then used to prove that the projective tensor cone of two arbitrary cones is a cone, and to find a new characterisation of finite-dimensional vector lattices.Comment: 8 page

Stochastic integration for Lévy processes with values in Banach spaces

AbstractA stochastic integral of Banach space valued deterministic functions with respect to Banach space valued Lévy processes is defined. There are no conditions on the Banach spaces or on the Lévy processes. The integral is defined analogously to the Pettis integral. The integrability of a function is characterized by means of a radonifying property of an integral operator associated with the integrand. The integral is used to prove a Lévy–Itô decomposition for Banach space valued Lévy processes and to study existence and uniqueness of solutions of stochastic Cauchy problems driven by Lévy processes

Invariant measures for stochastic cauchy problems with asymptotically unstable drift semigroup

We investigate existence and permanence properties of invariant measures for abstract stochastic Cauchy problems of the for

An analytical introduction to stochastic differential equations: Part I - the Langevin equation

We present an introduction to the theory of stochastic differential equations, motivating and explaining ideas from the point of view of analysis. First the notion of white noise is developed, introducing at the same time probabilistic tools. Then the one dimensional Langevin equation is formulated as a deterministic integral equation with a parameter. Its solution leads to stochastic convolution, which is defined as a Riemann-Stieltjes integral. It is shown that the parameter dependence yields a Gaussian system, of which the means and covariances arde computed. We conclude by introducing briefly the notion of invariant measure and the associated Kolmogorov equations

Invariant measures for stochastic cauchy problems with asymptotically unstable drift semigroup

We investigate existence and permanence properties of invariant measures for abstract stochastic Cauchy problems of the formDelft Institute of Applied MathematicsElectrical Engineering, Mathematics and Computer Scienc