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