We prove a new Burkholder–Rosenthal type inequality for discrete-time processes taking values in a 2-smooth Banach space. As a first application we prove that if \((S(t,s))_{0\leqslant s\le t\leqslant T}\) is a \(C_0\)-evolution family of contractions on a 2-smooth Banach space X and \((W_t)_{t\in [0,T]}\) is a cylindrical Brownian motion on a probability space \((\Omega ,{\mathbb {P}})\) adapted to some given filtration, then for every \(0<p<\infty \) there exists a constant \(C_{p,X}\) such that for all progressively measurable processes \(g: [0,T]\times \Omega \rightarrow X\) the process \((\int _0^t S(t,s)g_s\,\mathrm{d} W_s)_{t\in [0,T]}\) has a continuous modification and

Moreover, for \(2\leqslant p<\infty \) one may take \(C_{p,X} = 10 D \sqrt{p},\) where D is the constant in the definition of 2-smoothness for X. The order \(O(\sqrt{p})\) coincides with that of Burkholder’s inequality and is therefore optimal as \(p\rightarrow \infty \). Our result improves and unifies several existing maximal estimates and is even new in case X is a Hilbert space. Similar results are obtained if the driving martingale \(g_t\,\mathrm{d} W_t\) is replaced by more general X-valued martingales \(\,\mathrm{d} M_t\). Moreover, our methods allow for random evolution systems, a setting which appears to be completely new as far as maximal inequalities are concerned. As a second application, for a large class of time discretisation schemes (including splitting, implicit Euler, Crank-Nicholson, and other rational schemes) we obtain stability and pathwise uniform convergence of time discretisation schemes for solutions of linear SPDEs

where the family \((A(t))_{t\in [0,T]}\) is assumed to generate a \(C_0\)-evolution family \((S(t,s))_{0\leqslant s\leqslant t\leqslant T}\) of contractions on a 2-smooth Banach spaces X. Under spatial smoothness assumptions on the inhomogeneity g, contractivity is not needed and explicit decay rates are obtained. In the parabolic setting this sharpens several know estimates in the literature; beyond the parabolic setting this seems to provide the first systematic approach to pathwise uniform convergence to time discretisation schemes.

我们证明了一个新的 Burkholder-Rosenthal 型不等式，用于在 2-smooth Banach 空间中取值的离散时间过程。作为第一个应用，我们证明如果\((S(t,s))_{0\leqslant s\le t\leqslant T}\)是一个\(C_0\) - 2-smooth 上的收缩演化族的Banach空间X和（{在吨\ [0，T]} ...（W_T）_ \）\是在概率空间中的圆筒状的布朗运动\（（\欧米茄，{\ mathbb {P}}）\）适于一些给定过滤，然后对于每个\(0<p<\infty \)存在一个常数\(C_{p,X}\)使得对于所有渐进可测量的过程\(g: [0,T]\times \Omega \rightarrow X\)过程\((\int _0^t S(t,s)g_s\,\mathrm{d} W_s)_{t\in [0,T]}\)有一个连续的修改和

此外，对于\(2\leqslant p<\infty \)可以采用\(C_{p,X} = 10 D \sqrt{p},\)其中D是X的 2-平滑度定义中的常数. 顺序\(O(\sqrt{p})\)与伯克霍尔德不等式的顺序一致，因此最优为\(p\rightarrow \infty \)。我们的结果改进并统一了几个现有的最大估计，并且在X是希尔伯特空间的情况下甚至是新的。如果将驱动鞅\(g_t\,\mathrm{d} W_t\)替换为更一般的X值鞅\(\,\mathrm{d} M_t\). 此外，我们的方法允许随机进化系统，就最大不等式而言，这种设置似乎是全新的。作为第二个应用，对于一大类时间离散化方案（包括分裂、隐式欧拉、Crank-Nicholson 和其他有理方案），我们获得了线性 SPDE 解的时间离散化方案的稳定性和路径一致收敛

其中假设族\((A(t))_{t\in [0,T]}\)生成一个\(C_0\) -进化族\((S(t,s))_{0 \leqslant s\leqslant t\leqslant T}\)在 2 平滑 Banach 空间X上的收缩。在不均匀性g 的空间平滑假设下，不需要收缩性并且获得显式衰减率。在抛物线设置中，这提高了文献中的几个已知估计值；除了抛物线设置之外，这似乎提供了第一个系统化的路径统一收敛到时间离散化方案的方法。