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5), multiply both sides by G and take expectations. 9). 8) holds for multiindices of order k − 1. Fix α = (α1 , · · · , αk ). Then, E (∂α ϕ)(F )G = E ∂(α1 ,··· ,αk−1 ) ((∂αk ϕ)(F ))G = E (∂αk ϕ)(F )H(α1 ,··· ,αk−1 ) (F, G) = E ϕ(F )Hαk (F, H(α1 ,··· ,αk−1 ) (F, G) . The proof is complete. Comments The results of this chapter are either rephrasings or quotations of statements from [7] [18], [41], [46], [61], [65], just to mention a few of them. The common source is [32]. 51 7 Stochastic partial differential equations driven by a Gaussian spatially homogeneous correlated noise The purpose of the rest of this course is to apply the criteria established in Chapter 5 to random vectors which are solutions of SPDEs at fixed points.

Set ϕ (x) = ϕ( x ), > 0. Let x Ψ (x) = ϕ (y)dy. −∞ 30 The chain rule yields Ψ (F ) ∈ D1,1 and DΨ (F ) = ϕ (F )DF . Let u be an H-valued random variable of the type n u= F j hj , j=1 with Fj ∈ Sb . We notice that the duality relation between D and δ holds for F ∈ D1,1 ∩ L∞ (Ω) and u of the kind described before. Moreover, u is total in L1 (Ω; H) , that means if v ∈ L1 (Ω; H) satisfies E( v, u H ) = 0 for any u in the class, then v = 0. Then we have |E(ϕ (F ) DF, u Taking limits as H )| = |E( (D(Ψ (F )), u H )| = |E(Ψ (F )δ(u))| ≤ ||ϕ||∞ E(|δ(u)|).

F = W 2 (T ), 2. F = W 3 (T ), 3. F = (W (T ) + T ) exp(−W (T ) − 12 T ). 10), find the ˜ for the following random integral representation with respect to the integrator W variables: 1. F = W 2 (T ), θ deterministic, 2. F = exp( T 0 λ(s)dW (s)), λ, θ deterministic, 3. F = exp( T 0 λ(s)dW (s)), λ deterministic and θ(s) = W (s). 45 6 Criteria for absolute continuity and smoothness of probability laws In Chapter 1 we have given general results ensuring existence and smoothness of density of probability laws.