On the equivalence of solutions for a class of stochastic evolution equations in a Banach space

Abstract: We study a class of stochastic evolution equations in a Banach space $E$ driven by cylindrical Wiener process. Three different concept 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 equation with additive noise. We also consider two examples of these equations in non-reflexive Banach spaces: a stochastic transport equation with delay and a stochastic McKendrick equation with delay.