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Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem

Zdzisław Brzeźniak, Jan van Neerven (2000)

Studia Mathematica

Let H be a separable real Hilbert space and let E be a separable real Banach space. We develop a general theory of stochastic convolution of ℒ(H,E)-valued functions with respect to a cylindrical Wiener process W t H t [ 0 , T ] with Cameron-Martin space H. This theory is applied to obtain necessary and sufficient conditions for the existence of a weak solution of the stochastic abstract Cauchy problem (ACP) d X t = A X t d t + B d W t H (t∈ [0,T]), X 0 = 0 almost surely, where A is the generator of a C 0 -semigroup S ( t ) t 0 of bounded linear operators on...

Stochastic integration of functions with values in a Banach space

J. M. A. M. van Neerven, L. Weis (2005)

Studia Mathematica

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...

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