Abstract
Let E be a type 2 umd Banach space, H a Hilbert space and let p∈[1,∞). Consider the following stochastic delay equation in E:
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).
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Communicated by Markus Haase.
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Cox, S., Górajski, M. Vector-valued stochastic delay equations—a semigroup approach. Semigroup Forum 82, 389–411 (2011). https://doi.org/10.1007/s00233-010-9276-4
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DOI: https://doi.org/10.1007/s00233-010-9276-4