A mild Itô formula for SPDEs
نویسندگان
چکیده
This article introduces a new somehow mild Itô type formula for the solution process of a stochastic partial differential equation of evolutionary type. 1 A mild Itô formula for SPDEs Throughout this article suppose that the following setting and the following assumptions are fulfilled. Fix T ∈ (0,∞) and t0 ∈ [0, T ), let (Ω,F ,P) be a probability space with a normal filtration (Ft)t∈[t0,T ] and let (H, ⟨·, ·⟩H , ∥·∥H) and (U, ⟨·, ·⟩U , ∥·∥U) be two separable R-Hilbert spaces. In addition, let Q : U → U be a bounded nonnegative symmetric linear operator and let (Wt)t∈[t0,T ] be a cylindrical Q-Wiener process with respect to (Ft)t∈[t0,T ]. Assumption 1 (Linear operator A). Let I be a finite or countable set and let (λi)i∈I ⊂ R be a family of real numbers with infi∈I λi > −∞. Moreover, let (ei)i∈I ⊂ H be an orthonormal basis of H and let A : D(A) ⊂ H → H be a linear operator with Av = ∑ i∈I −λi ⟨ei, v⟩H ei (1) for all v ∈ D(A) and with D(A) = { w ∈ H ∣∣∑ i∈I |λi| 2 |⟨ei, w⟩H | 2 < ∞ } . Let η ∈ [0,∞) be a nonnegative real number with η > − infi∈I λi. By ( Hr := D ((η − A)) , ⟨·, ·⟩Hr , ∥·∥Hr ) for r ∈ R we denote the R-Hilbert spaces of domains of fractional powers of the linear operator η−A : D(A) ⊂ H → H. Assumption 2 (Drift term F ). Let α, γ ∈ R be real numbers with γ−α < 1 and let F : Hγ → Hα be globally Lipschitz continuous.
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تاریخ انتشار 2010