Stochastic integration of functions with values in a Banach space

• Autorzy: J. M. A. M. van Neerven , L. Weis
• Języki publikacji: EN

Abstrakty

EN

Let H be a separable real Hilbert space and let E be a real Banach space. In this paper we construct a stochastic integral for certain operator-valued functions Φ: (0,T) → ℒ(H,E) with respect to a cylindrical Wiener process ${W_H(t)}_{t∈[0,T]}$. The construction of the integral is given by a series expansion in terms of the stochastic integrals for certain E-valued functions. As a substitute for the Itô isometry we show that the square expectation of the integral equals the radonifying norm of an operator which is canonically associated with the integrand. We obtain characterizations for the class of stochastically integrable functions and prove various convergence theorems. The results are applied to the study of linear evolution equations with additive cylindrical noise in general Banach spaces. An example is presented of a linear evolution equation driven by a one-dimensional Brownian motion which has no weak solution.

Kategorie tematyczne

• 47D06: One-parameter semigroups and linear evolution equations
• 35R15: Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables)
• 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
• 60H15: Stochastic partial differential equations
• 60H05: Stochastic integrals

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2005
• Tom: 166
• Numer: 2
• Strony: 131-170

Daty

wydano

2005

Twórcy

autor

J. M. A. M. van Neerven

• Delft Institute of Applied Mathematics, Technical University of Delft, P.O. Box 5031, 2600 GA Delft, The Netherlands

autor

L. Weis

• Mathematisches Institut I, Technische Universität Karlsruhe, D-76128 Karlsruhe, Germany

Identyfikatory

DOI

10.4064/sm166-2-2