In this article we introduce a stochastic counterpart of the Hörmander condition and Calderón-Zygmund theorem. Let Wt be a Wiener process in a probability space Ω and let K(ω, r, t, x, y) be a random kernel which is allowed to be stochastically singular in a domain O ⊂ Rd in the sense that E______t 0_|x-y|<ϵ |K(ω, s, t, y, x)|dydWs_____p =∞ ∀t, p, ϵ > 0, x ∈ O. We prove that the stochastic integral operator of the type Tg(t, x) :=_t 0_O (0.1) K(ω, s, t, y, x)g(s, y)dydWs is bounded on Lp = Lp (Ω × (0,∞);Lp(O)) for all p ∈ [2,∞) if it is bounded on L2 and the following (which we call stochastic Hörmander condition) holds: There exists a quasi-metric ρ on (0,∞)× O and a positive constant C0 such that for X = (t, x), Y = (s, y), Z = (r, z) ∈ (0,∞)×O, sup ω∈Ω,X,Y_∞ 0__ρ(X,Z)≥C0ρ(X,Y ) |K(r, t, z, x)-K(r, s, z, y)| dz_2 dr < ∞. Such a stochastic singular integral naturally appears when one proves the maximal regularity of solutions to stochastic partial differential equations (SPDEs). As applications, we obtain the sharp Lp-regularity result for a wide class of SPDEs, which includes SPDEs with time measurable pseudo-differential operators and SPDEs defined on non-smooth angular domains.
ASJC Scopus subject areas
- Applied Mathematics