1. Department of Applied Mathematical Analysis, Delft University of Technology, P.O. Box 5031, 2600, GA Delft, The Netherlands 2. Mathematisches Institut I, Technische Universit?t Karlsruhe, 76128, Karlsruhe, Germany
Abstract:
Let E be a real Banach space with property (α) and let WΓ be an E-valued Brownian motion with distribution Γ. We show that a function is stochastically integrable with respect to WΓ if and only if Γ-almost all orbits Ψx are stochastically integrable with respect to a real Brownian motion. This result is derived from an abstract result on existence
of Γ-measurable linear extensions of γ-radonifying operators with values in spaces of γ-radonifying operators. As an application we obtain a necessary and sufficient condition for solvability of stochastic evolution
equations driven by an E-valued Brownian motion.
The first named author gratefully acknowledges the support by a ‘VIDI subsidie’ in the ‘Vernieuwingsimpuls’ programme of The
Netherlands Organization for Scientific Research (NWO) and the Research Training Network HPRN-CT-2002–00281. The second named
author was supported by grants from the Volkswagenstiftung (I/78593) and the Deutsche Forschungsgemeinschaft (We 2847/1–1).