By G. E. H Reuter
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Extra resources for Elementary differential equations and operators,
S. 2) Fn = F on Ωn . For F ∈ Dk,p loc (V ), we deﬁne DF = DFn , on Ωn . The local property of D ensures that this is well deﬁned. Analogously, if u is an element 1,2 of D1,2 loc (H), we deﬁne δ(u) = δ(un ) on Ωn . Remember that D (H) ⊂ Dom δ. Comments This chapter requires knowledge of Itˆ o’s results on multiple stochastic integrals and their role in the Wiener chaos decomposition. They are proved in reference . One could entitle this chapter “Essentials of Malliavin Calculus”. Indeed, it is a very brief account of a deep and large theory presented in a quite simpliﬁed way.
Therefore, the action of the operators and their domains can be described in terms of conditions on these integrals. We gain in operativeness because additional tools of stochastic calculus become available. For the sake of completeness we start with a very short account on multiple Itˆ o-Wiener integrals and their role in the Wiener chaos decomposition. For complete details on the topic we refer the reader to the original work by Itˆ o  (see also ref. ). The framework here consists of a separable σ-ﬁnite measure space (A, A, m), the Hilbert space H = L2 (A, A, m) and the white noise W = (W (F ), F ∈ A) based on m.
F˜n−1 , f m(dt) L2 (An ) = E In (f˜n−1 )In (f ) = E In (f˜n−1 )F . Assume that u belongs to Dom δ. The preceding equalities show that on the Wiener chaos of order n, n ≥ 1, δ(u) = In (f˜n−1 ). 33) holds true. 32) converges. Then, by the arguments above, we obtain ∞ E u, DF H ≤ F In (f˜n−1 ) 2 n=1 ≤C F 2. 33) holds. 21) can be extended to random vectors u ∈ D2,2 (H). 21) holds in the less restrictive situation u ∈ D1,2 (H). This is the goal of the next statement. 15 Let u ∈ D1,2 (H). Assume that for almost every t ∈ A, the process (Dt u(s), s ∈ A) belongs to Dom δ and there is a version of the process δ(Dt u(s)), t ∈ A which is in L2 (Ω × A).