J. M. A. M. van Neerven

Stochastic integration in UMD Banach spaces

In this paper we develop a stochastic integration theory for processes with values in a quasi-Banach space. The integrator is a cylindrical Brownian motion. The main results give sufficient conditions for stochastic integrability. They are natural extensions of known results in the Banach space setting. We apply our main results to the stochastic heat equation where the forcing terms are assumed to have Besov regularity in the space variable with integrability exponent

Itô's formula in UMD Banach spaces and regularity of solutions of the Zakai equation

p\in (0,1]

Transition Semigroups of Banach Space-Valued Ornstein–Uhlenbeck Processes

. The latter is natural to consider for its potential application to adaptive wavelet methods for stochastic partial differential equations.

Invariant measures for the linear stochastic Cauchy problem and R-boundedness of the resolvent

Using the theory of stochastic integration for processes with values in a UMD Banach space developed recently by the authors, an Ito formula is proved which is applied to prove the existence of strong solutions for a class of stochastic evolution equations in UMD Banach spaces. The abstract results are applied to prove regularity in space and time of the solutions of the Zakai equation.

Lower semicontinuity and the theorem of Datko and Pazy

We investigate the transition semigroup of the solution to a stochastic evolution equation dX(t)=AX(t) dt+dWH(t), t≥0, where A is the generator of a C0-semigroup S on a separable real Banach space E and WH(t)t≥0 is cylindrical white noise with values in a real Hilbert space H which is continuously embedded in E. Various properties of these semigroups, such as the strong Feller property, the spectral gap property, and analyticity, are characterized in terms of the behaviour of S in H. In particular we investigate the interplay between analyticity of the transition semigroup, S-invariance of H, and analyticity of the restricted semigroup SH.

On Closability of Directional Gradients

Abstract.We study the asymptotic behaviour of solutions of the stochastic abstract Cauchy problem

Uniqueness of invariant measures for the stochastic Cauchy problem in Banach spaces

\left\{ {\begin{array}{*{20}l} {dU\left( t \right) = AU\left( t \right)dt + BdW_H \left( t \right),\quad t \geqslant 0,} \hfill\\ {U\left( 0 \right) = 0,} \hfill\\ \end{array}} \right.

Approximating Bochner integrals by Riemann sums

where A is the generator of a C0-semigroup on a Banach space E, WH is a cylindrical Brownian motion over a separable Hilbert space H, and