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


 
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1. Title Title of document A maximal inequality for stochastic convolutions in 2-smooth Banach spaces
 
2. Creator Author's name, affiliation, country Jan Van Neerven; Delft University of Techonology; Netherlands
 
2. Creator Author's name, affiliation, country Jiahui Zhu; Delft University of Techonology; Netherlands
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Stochastic convolutions, maximal inequality, $2$-smooth Banach spaces, It^o formula.
 
3. Subject Subject classification Primary 60H05; Secondary 60H15
 
4. Description Abstract

Let $(e^{tA})_{t\geq0}$ be a $C_0$-contraction semigroup on a $2$-smooth Banach space $E$, let $(W_t)_{t\geq0}$ be a cylindrical Brownian motion in a Hilbert space $H$, and let $(g_t)_{t\geq0}$ be a progressively measurable process with values in the space $\gamma(H,E)$ of all $\gamma$-radonifying operators from $H$ to $E$. We prove that for all $0<p<\infty$ there exists a constant $C$, depending only on $p$and $E$, such that for all $T\geq0$ we have $$E\sup_{0\leq t\leq T}\left\Vert\int_0^t\!e^{(t-s)A}\,g_sdW_s\right\Vert^p\leq CE\left(\int_0^T\!\left(\left\Vert g_t\right\Vert_{\gamma(H,E)}\right)^2\,dt\right)^{p/2}.$$ For $p\geq2$ the proof is based on the observation that $\psi(x)=\Vert x\Vert^p$ is Fréchet differentiable and its derivative satisfies the Lipschitz estimate $\Vert \psi'(x)-\psi'(y)\Vert\leq C\left(\Vert x\Vert+\Vert y\Vert\right)^{p-2}\Vert x-y\Vert$; the extension to $0<p<2$ proceeds via Lenglart’s inequality.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) VICI subsidy 639.033.604 of the Netherlands Organisation for Scientific Research (NWO)
 
7. Date (YYYY-MM-DD) 2011-11-20
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1677
 
10. Identifier Digital Object Identifier 10.1214/ECP.v16-1677
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 16
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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