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

2011/05/24 by Jan van Neerven, Jiahui Zhu, van Neerven, Jan +1
Mathematics · #60H05 (Primary) 60H15 (Secondary) #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR) #math.FA #math.PR #msc:60H05 #msc:60H15

paper · pdf · doi:10.48550/arxiv.1105.4720

Minor revisions and example added. Accepted for publication in Electron. Commun. Probab

arxiv created 2011/10/25 · arxiv updated 2011/10/26

Abstract

Let (e^tA)_{t ≥ 0} be a C₀-contraction semigroup on a 2-smooth Banach space E, let (W_t)_{t ≥ 0} be a cylindrical Brownian motion in a Hilbert space H, and let (g_t)_{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} || ∫₀^t e^(t-s)A g_s dW_s ||^p ≤ C 𝔼 (∫₀^T || g_t ||_γ(H,E)² dt)^(p)/(2). For p ≥ 2 the proof is based on the observation that ψ(x) = || x ||^p is Fréchet differentiable and its derivative satisfies the Lipschitz estimate || ψ'(x) - ψ'(y)|| ≤ C(|| x || + || y ||)^p-2 || x-y ||; the extension to 0<p<2 proceeds via Lenglart's inequality.

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