Abstract
Let E be a real Banach space with property (α) and let W Γ be an E-valued Brownian motion with distribution Γ. We show that a function \(\Psi:[0,T]\to{\mathcal L}(E)\) is stochastically integrable with respect to W Γ 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.
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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).
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van Neerven, J.M.A.M., Weis, L. Stochastic Integration of Operator-Valued Functions with Respect to Banach Space-Valued Brownian Motion. Potential Anal 29, 65–88 (2008). https://doi.org/10.1007/s11118-008-9088-2
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DOI: https://doi.org/10.1007/s11118-008-9088-2
Keywords
- Stochastic integration in Banach spaces
- γ-Radonifying operators
- Property(α)
- Measurable linear extensions
- Stochastic evolution equations