Infinitely delayed stochastic evolution equations on UMD Banach spaces

We prove existence and uniqueness of solutions for a class of infinitely delayed stochastic evolution equations with multiplicative noise term { d U ( t ) = ( A U ( t ) + F ( t , U t ) ) d t + G ( t , U t ) d W ( t ) , t ∈ [ 0 , T 0 ] , U ( 0 ) = X : Ω → E , U 0 = Φ : ( − ∞ , 0 ] × Ω → E , where A is the generator of an analytic semigroup on a UMD Banach space E and F and G are functions from the history of the system satisfying Lipschitz conditions. This paper is based on recent work of van Neerven et al., developing the theory of abstract stochastic evolution equations in UMD Banach spaces.