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

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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) $dX_t = AX_tdt + BdW_t^H$ (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 E and B ∈ ℒ(H,E) is a bounded linear operator. We further show that whenever a weak solution exists, it is unique, and given by a stochastic convolution $X_t = ∫^{t}_{0} S(t-s)BdW_{s}^{H}$.
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  • Department of Mathematics, The University of Hull, Hull HU6 7RX, England
  • Department of Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands
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