Electronic Communications in Probability

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

Jan Van Neerven and Jiahui Zhu

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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.

Article information

Electron. Commun. Probab., Volume 16 (2011), paper no. 60, 689-705.

Accepted: 20 November 2011
First available in Project Euclid: 7 June 2016

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Zentralblatt MATH identifier

Primary: 60H05: Stochastic integrals
Secondary: 60H15: Stochastic partial differential equations [See also 35R60]

Stochastic convolutions maximal inequality $2$-smooth Banach spaces It^o formula

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Van Neerven, Jan; Zhu, Jiahui. A maximal inequality for stochastic convolutions in 2-smooth Banach spaces. Electron. Commun. Probab. 16 (2011), paper no. 60, 689--705. doi:10.1214/ECP.v16-1677. https://projecteuclid.org/euclid.ecp/1465262016

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