A maximal inequality for stochastic convolutions in 2-smooth Banach spaces

Abstract

Let (e tA ) t 0 be a C0-contraction semigroup on a 2-smooth Banach space E, let (Wt) t 0 be a cylindrical Brownian motion in a Hilbert space H, and let (gt) t 0 be a progressively measurable process with values in the space γ(H, E) of all γ-radonifying operators from H to E. We prove that for all 0 < p < ∞ there exists a constant C, depending only on p and E, such that for all T 0 we have E sup 0 t T t 0 e (t−s)A gs dWs p

Well-posedness for a Class of Dissipative Stochastic Evolution Equations with Wiener and Poisson Noise

“…for all t ∈ [0, T ], and that w ε λ = (I + εA) −1 v λ → v λ in H p as ε → 0. We are going to apply Itô's formula (in particular we shall use the version in [5,Thm. 3.1]) to obtain estimates for w ε λ p E .…”

“…where R is a "remainder" term, the precise definition of which is given in [5]. Note that ψ(u) = u 2 .…”

A version of the Hörmander–Malliavin theorem in 2-smooth Banach spaces

“…We obtain that Y t y verifies (17). But the solution to (17) is unique in S 2 (E). From this and from the continuity of paths it follows that for all t ∈ [0, T ] Y t y = ξ t a.s.…”

Finite-time blow-up of a non-local stochastic parabolic problem

“…The choice of this scale of function spaces is more natural for our method than the Sobolev spaces W r p (D). Let 1 < r ≤ 2 then we say that a Banach space X is r-smooth if the modulus of smoothness [26] ρ · (t) = sup…”

“…satisfies ρ · (t) ≤ Ct r , for all t > 0, see [26]. Let K be a separable Hilbert space and X be a 2-smooth Banach space.…”