Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemes
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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≤ s≤ T is a C_0-evolution family of contractions on a 2-smooth Banach space X and (W_t)_t∈ [0,T] is a cylindrical Brownian motion on a probability space (Ω,P), then for every 0<p<∞ there exists a constant C_p,X such that for all progressively measurable processes g: [0,T]×Ω→ X the process (∫_0^t S(t,s)g_sdW_s)_t∈ [0,T] has a continuous modification and Esup_t∈ [0,T]∫_0^t S(t,s)g_sdW_s ^p≤ C_p,X^p 𝔼(∫_0^T g_t^2_γ(H,X)dt)^p/2. Moreover, for 2≤ p<∞ one may take C_p,X = 10 D √(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 g_tdW_t is replaced by more general X-valued martingales dM_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 we obtain stability and pathwise uniform convergence of time discretisation schemes for solutions of linear SPDEs du_t = A(t)u_tdt + g_tdW_t, 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.
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