Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemes

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 $(S(t,s)){0\leq s\leq T}$ is a $C0$-evolution family of contractions on a $2$-smooth Banach space $X$ and $(Wt){t\in [0,T]}$ is a cylindrical Brownian motion on a probability space $(\Omega,P)$, then for every $0<p<\infty$ there exists a constant $C{p,X}$ such that for all progressively measurable processes $g: [0,T]\times \Omega\to X$ the process $(\int0^t S(t,s)gsdWs){t\in [0,T]}$ has a continuous modification and $$E\sup{t\in [0,T]}\Big| \int0^t S(t,s)gsdWs \Big|^p\leq C{p,X}^p \mathbb{E} \Bigl(\int0^T | gt|^{2_{\gamma(H,X)}dt\Bigr){p/2}.$$} Moreover, for $2\leq p<\infty$ one may take $C{p,X} = 10 D \sqrt{p},$ where $D$ is the constant in the definition of $2$-smoothness for $X$. 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 $gtdWt$ is replaced by more general $X$-valued martingales $dMt$. 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 we obtain stability and pathwise uniform convergence of time discretisation schemes for solutions of linear SPDEs $$ dut = A(t)utdt + gtdWt, \quad u_0 = 0,$$ 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.