Function Space-Time Manifolds by Michal A.D.

By Michal A.D.

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7) always holds. 3 is an immediate consequence of the celebrated Hille-Yosida Theorem. The proof of (i) is contained in Pazy [1] and part (ii) can be found in Hille and Phillips [1]. We shall not go into further details here because these results are well-established. 5) is generally not true in infinite dimensional spaces, a fact which is illustrated by the example below. 2 For a measurable function f defined on [0, ∞), set ∞ f exp = es |f (s)|ds, 0 and let E be the space of all measurable functions f (·) on [0, ∞) for which f exp < ∞.

Suppose p ≥ 2, 0 ≤ t ≤ T , and let Lpt (Ω, F, P ; H), simply Lpt (Ω; H), be the subspace of Lp (Ω, F, P ; H) which consists of all Ft -measurable random variables. For arbitrary 0 ≤ s ≤ T , let Ca ([s, T ]; Lp (Ω, F, P ; H)) be the subspace of C([s, T ]; Lp (Ω, F, P ; H)) which consists of {Ft }-adapted processes. 12), however, with initial datum xs ∈ Lps (Ω; H), s ≤ t ≤ T , t Xt = T (t − s)xs + t T (t − u)F (u, Xu )du + s Xs = xs ∈ Lps (Ω; H). 4 in the following form. 5 For any 0 ≤ s ≤ t ≤ T , there exists a unique map U (t, s) : Lps (Ω; H) → Lpt (Ω; H) with properties: Stochastic Differential Equations in Infinite Dimensions 29 (i).

Xt is continuous in t ∈ I almost surely. For arbitrary 0 ≤ t ≤ T , t P ω : 0 Xs (ω) 2H ds < ∞ = 1 and t Xt = x0 + t (AXs + F (s, Xs ))ds + 0 for any x0 ∈ D(A) almost surely. 14) 24 Stability of Infinite Dimensional Stochastic Differential Equations By a straightforward argument, it is possible to establish the following result. 10). 10) is unique. The following stochastic version of the classic Fubini theorem will be frequently used in the book and its proof can be found in Da Prato and Zabczyk [1].

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