caa a22 4500 445344814 CHVBK 20180317142825.0 cr unu---uuuuu 170323e20110601xx s 000 0 eng 10.1007/s00233-010-9276-4 doi (NATIONALLICENCE)springer-10.1007/s00233-010-9276-4 Vector-valued stochastic delay equations—asemigroup approach [Elektronische Daten] [Sonja Cox, Mariusz Górajski] Let E be a type 2 umd Banach space, H a Hilbert space and let p∈[1,∞). Consider the following stochastic delay equation inE: SDE$$ \left\{\begin{array}{l}dX(t)= AX(t) + C X_t + B(X(t),X_t)dW_H(t),\quad t>0;\\[5pt]X(0)=x_0;\\[5pt]X_0=f_0,\end{array}\right. $$ where A:D(A)⊂E→E is the generator of a C 0-semigroup. The operator $C\in\mathcal{L}(W^{1,p}(-1,0;E),E)$ is given by a Riemann-Stieltjes integral, and B:E×L p(−1,0;E)→γ(H,E) is a Lipschitz function. Moreover W H is an H-cylindrical Brownian motion adapted to and , . We prove that a solution to (SDE) is equivalent to a solution to the corresponding stochastic Cauchy problem, and use this to prove the existence, uniqueness and continuity of a solution to(SDE). The Author(s), 2010 Stochastic partial differential equations with finite delay nationallicence Stochastic Cauchy problem nationallicence umd Banach spaces nationallicence Type 2 Banach spaces nationallicence Cox Sonja Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA, Delft, The Netherlands aut Górajski Mariusz Faculty of Mathematics and Computer Science, University of Łódź, Banacha 22, 90-238, Łódź, Poland aut Semigroup Forum Springer-Verlag 82/3(2011-06-01), 389-411 0037-1912 82:3<389 2011 82 233 https://doi.org/10.1007/s00233-010-9276-4 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00233-010-9276-4 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Cox Sonja Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA, Delft, The Netherlands aut NATIONALLICENCE 700 1- Górajski Mariusz Faculty of Mathematics and Computer Science, University of Łódź, Banacha 22, 90-238, Łódź, Poland aut NATIONALLICENCE 773 0- Semigroup Forum Springer-Verlag 82/3(2011-06-01), 389-411 0037-1912 82:3<389 2011 82 233 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer