Abstract
Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions
Highlights
Let H be a separable Hilbert space, with inner product ·, · H and norm · H
Let (U, U ) be another separable Hilbert space and let L2(U, H) denote the space of all Hilbert-Schmidt operators from U to H equipped with the usual Hilbert-Schmidt norm L2
Over the few lemmas we carefully develop a pathwise definition of the stochastic integral
Summary
Let H be a separable Hilbert space, with inner product ·, · H and norm · H. A pair (X, W ), where X = (X(t))t∈[0,∞) is an (Ft)-adapted process with paths in B and W is a standard R∞-Wiener process on a stochastic basis (Ω, F , P, (Ft)) is called a weak solution of (1.1) if (i) For any T ∈ [0, ∞). To define strong solutions we need to introduce the following class Eof maps: Let Edenote the set of all maps F : H × W0 → B such that for every probability measure μ on (H, B(H)). For every standard R∞-Wiener process on a stochastic basis (Ω, F , P, (Ft)) and any F0/B(H)measurable ξ : Ω → H the continuous process.
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