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

Mariusz Górajski

doi:10.1007/s00020-013-2119-4

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.

Highlights

Let E be a Banach space and let H be a separable Hilbert space
The additional condition in the definition of analytically strong solution is not appropriate for stochastic delay equations which we consider in Sect. 5, in this paper we do not focus on this concept of solution
In this paper we prove that the equivalence of Definitions 1.1–1.3 is valid in umd− Banach spaces

Summary

Introduction

Let E be a Banach space and let H be a separable Hilbert space. In a given probability basis ((Ω, F, F, P), WH ), i.e. (Ω, F, P) is a complete probability space and WH is an H-cylindrical Wiener process with respect to a complete filtration F = (Ft)t≥0 on (Ω, F , P), consider the stochastic evolution equation: dY (t) = AY (t)dt + F (Y (t))dt + G(Y (t))dWH (t), Y (0) = Y0, t ≥ 0;. The additional condition in the definition of analytically strong solution is not appropriate for stochastic delay equations (see Remark 4.10 in [7]) which we consider, in this paper we do not focus on this concept of solution The equivalence of these three interpretations of solution to (SCP) in Hilbert space has been proved by Chojnowska-Michalik in [6] (see [11, Theorem 6.5] and [27, Theorem 9.15]). It is worth mentioning that it turns out that for stochastic evolution equations with non-additive noise in a umd Banach space which are not type 2 it is convenient to analyse a concept of mild Eη-solution of (SCP) This interpretation is more general than these considered in the article. Mainly based on [25], we present sufficient conditions for the existence of stochastic integral in a umd− Banach space, and some preliminary lemmas which will be useful in the sequel

Preliminaries

Equivalence of Weak and Mild Solutions

Stochastic Evolution Equation with Additive Noise

Stochastic Delay Evolution Equation

Examples