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

Abstract

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))_{0leqslant sle tleqslant T} is a C_0-evolution family of contractions on a 2-smooth Banach space X and (W_t)_{tin [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)_{tin [0,T]} has a continuous modification and Esupt∈[0,T]‖∫0tS(t,s)gsdWs‖p⩽Cp,XpE(∫0T‖gt‖γ(H,X)2dt)p/2.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathbb {E}}\\sup _{t\\in [0,T]}\\Big \\Vert \\int _0^t S(t,s)g_s\\,\\mathrm{d} W_s \\Big \\Vert ^p\\leqslant C_{p,X}^p {\\mathbb {E}} \\Bigl (\\int _0^T \\Vert g_t\\Vert ^2_{\\gamma (H,X)}\\,\\mathrm{d} t\\Bigr )^{p/2}. \\end{aligned}$$\\end{document}Moreover, for 2leqslant 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 prightarrow 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 dut=A(t)utdt+gtdWt,u0=0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\,\\mathrm{d} u_t = A(t)u_t\\,\\mathrm{d} t + g_t\\,\\mathrm{d} W_t, \\quad u_0 = 0, \\end{aligned}$$\\end{document}where the family (A(t))_{tin [0,T]} is assumed to generate a C_0-evolution family (S(t,s))_{0leqslant sleqslant tleqslant 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.