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On the equivalence of solutions for a class of stochastic evolution equations in a Banach space. (English) Zbl 1305.34136

Summary: We study a class of stochastic evolution equations in a Banach space \(E\) driven by cylindrical Wiener process. Three different analytical concepts of solutions: generalised strong, weak and mild are defined and the conditions under which they are equivalent are given. We apply this result to prove existence, uniqueness and continuity of weak solutions to stochastic delay evolution equations. We also consider two examples of these equations in non-reflexive Banach spaces: a stochastic transport equation with delay and a stochastic delay McKendrick equation.

MSC:

34K50 Stochastic functional-differential equations
60H30 Applications of stochastic analysis (to PDEs, etc.)
47D06 One-parameter semigroups and linear evolution equations
34K30 Functional-differential equations in abstract spaces
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