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

Abstract

Abstract. 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.Mathematics Subject Classification (2010). Primary 34K50; Secondary 60H30, 47D06.

Stochastic integration and stochastic PDEs driven by jumps on the dual of a nuclear space

“…For our proof we benefit from ideas taken from Peszat and Zabczyk [31] and Gorajski [16] on the proof of equivalence between weak and mild solutions on separable Hilbert spaces and in UMD Banach spaces respectively.…”

Vector-valued stochastic delay equations—A weak solution and its Markovian representation

“…In the introduction we have mentioned that the stochastic delay equation (SDE) can be rewritten as a stochastic Cauchy problem. In this section we recall the result concerning different concept of solution to (SCP) form [16]. Let E be a Banach space and H be a separable Hilbert space, and let A :…”

“…In the introduction we have mentioned that the stochastic delay equation (SDE) can be rewritten as a stochastic Cauchy problem. In this section we recall the result concerning different concept of solution to (SCP) form [16]. Let E be a Banach space and H be a separable Hilbert space, and let A : D(A) ⊂ E → E be the generator of a C 0 -semigroup (T (t)) t≥0 on E. The sun dual semigroup (T ⊙ (t)) t≥0 defined as subspace semigroup by T ⊙ (t) = T * (t) |E ⊙ defined on E ⊙ = D(A * ) is strongly continuous (see Section 2.6 in [14] and Chapter 1 in [24]).…”

“…The equivalence of mild, weak and generalised strong solution to (SCP) has been established in [16]. First we recall the hypotheses (HA') Assume that (HA) is satisfied and for all t > 0 and all g ∈ L 1 (0, t; E) the function 16]). Assume that E has umd − property and conditions (HA') and (HB) are satisfied.…”