Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem
Studia Mathematica, Tome 143 (2000) no. 1, pp. 43-74
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 $X_t = ∫^{t}_{0} S(t-s)BdW_{s}^{H}$.
@article{10_4064_sm_143_1_43_74,
author = {Zdzis{\l}aw Brze\'zniak and Jan van Neerven},
title = {Stochastic convolution in separable {Banach} spaces and the stochastic linear {Cauchy} problem},
journal = {Studia Mathematica},
pages = {43--74},
year = {2000},
volume = {143},
number = {1},
doi = {10.4064/sm-143-1-43-74},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/sm-143-1-43-74/}
}
TY - JOUR AU - Zdzisław Brzeźniak AU - Jan van Neerven TI - Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem JO - Studia Mathematica PY - 2000 SP - 43 EP - 74 VL - 143 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.4064/sm-143-1-43-74/ DO - 10.4064/sm-143-1-43-74 LA - en ID - 10_4064_sm_143_1_43_74 ER -
%0 Journal Article %A Zdzisław Brzeźniak %A Jan van Neerven %T Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem %J Studia Mathematica %D 2000 %P 43-74 %V 143 %N 1 %U http://geodesic.mathdoc.fr/articles/10.4064/sm-143-1-43-74/ %R 10.4064/sm-143-1-43-74 %G en %F 10_4064_sm_143_1_43_74
Zdzisław Brzeźniak; Jan van Neerven. Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem. Studia Mathematica, Tome 143 (2000) no. 1, pp. 43-74. doi: 10.4064/sm-143-1-43-74
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