Infinitely delayed stochastic evolution equations on UMD Banach spaces

Abstract

We prove existence and uniqueness of solutions for a class of infinitely delayed stochastic evolution equations with multiplicative noise termwhere 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.

Stochastic Integration in Banach Spaces – a Survey

“…Applications of the theory of stochastic integration in UMD spaces have been worked out in a number of papers; see [12,13,18,21,22,24,30,57,56,58,78,80,81,93,96] and the references therein. Here we will limit ourselves to the maximal regularity theorem for stochastic convolutions from [81] which is obtained by combining Theorem 7.1 and 7.3 below, and which crucially depends on the sharp two-sided inequality of Theorem 5.5.…”

On the Equivalence of Solutions for a Class of Stochastic Evolution Equations in a Banach Space

“…Here we consider delay evolution equation with the Wiener additive noise, for delay equation in umd type 2 Banach space with more general Wiener noises we refer to [7], [16], where the reader can also find a more extensive literature overview. For stochastic delay evolution equation with infinite delay see [9]. At the end of this section we give two examples of stochastic delay partial differential equation in non-reflexive Banach space.…”

“…What is left to show is that [0, t] ∋ u → T (u)[ψ, 0] ′ represents an operator [R π1 , R π2 ] ′ in γ(L 2 (0, t; H), E p ) if and only if condition (Hψ) holds. Indeed, by the properties of delay semigroup (T (t)) t≥0 (see (9) in [16] and Proposition 3.11 in [3]) we have π 1 T (u)[ψ, 0] ′ = S(u)ψ and (π 2 T (u)[ψ, 0] ′ )(θ) = 1 (u+θ>0) S(u + θ)ψ for every u > 0 and a.e. θ ∈ [−1, 0].…”