A maximal inequality for stochastic convolutions in 2-smooth Banach spaces. (English) Zbl 1254.60053

van Neerven, Jan; Zhu, Jiahui

Electron. Commun. Probab. 16, 689-705 (2011).

Summary: 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 \[ \operatorname{E}\sup_{0\leq t\leq T}\left\|\int_0^t\!e^{(t-s)A}\,g_sdW_s\right\|^p\leq C\operatorname{E}\left(\int_0^T\!\left(\left\| g_t\right\|_{\gamma(H,E)}\right)^2\,dt\right)^{p/2}. \] For \(p\geq2\), the proof is based on the observation that \(\psi(x)=\| x\|^p\) is Fréchet differentiable and its derivative satisfies the Lipschitz estimate \(\| \psi'(x)-\psi'(y)\|\leq C\left(\| x\|+\| y\|\right)^{p-2}\| x-y\|\); the extension to \(0<p<2\) proceeds via Lenglart’s inequality.

MSC:

60H05	Stochastic integrals
60H15	Stochastic partial differential equations (aspects of stochastic analysis)

Keywords:

stochastic convolutions; maximal inequality; 2-smooth Banach spaces; Itô formula