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

2021; Springer Science+Business Media; Volume: 10; Issue: 2 Linguagem: Inglês

10.1007/s40072-021-00204-y

2194-041X

Jan van Neerven, Mark Veraar,

Advanced Harmonic Analysis Research

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 $$(S(t,s))_{0\leqslant s\le t\leqslant T}$$ ( S ( t , s ) ) 0 ⩽ s ≤ t ⩽ T is a $$C_0$$ C 0 -evolution family of contractions on a 2-smooth Banach space X and $$(W_t)_{t\in [0,T]}$$ ( W t ) t ∈ [ 0 , T ] is a cylindrical Brownian motion on a probability space $$(\Omega ,{\mathbb {P}})$$ ( Ω , P ) adapted to some given filtration, then for every $$0<p<\infty $$ 0 < p < ∞ there exists a constant $$C_{p,X}$$ C p , X such that for all progressively measurable processes $$g: [0,T]\times \Omega \rightarrow X$$ g : [ 0 , T ] × Ω → X the process $$(\int _0^t S(t,s)g_s\,\mathrm{d} W_s)_{t\in [0,T]}$$ ( ∫ 0 t S ( t , s ) g s d W s ) t ∈ [ 0 , T ] has a continuous modification and $$\begin{aligned} {\mathbb {E}}\sup _{t\in [0,T]}\Big \Vert \int _0^t S(t,s)g_s\,\mathrm{d} W_s \Big \Vert ^p\leqslant C_{p,X}^p {\mathbb {E}} \Bigl (\int _0^T \Vert g_t\Vert ^2_{\gamma (H,X)}\,\mathrm{d} t\Bigr )^{p/2}. \end{aligned}$$ E sup t ∈ [ 0 , T ] ‖ ∫ 0 t S ( t , s ) g s d W s ‖ p ⩽ C p , X p E ( ∫ 0 T ‖ g t ‖ γ ( H , X ) 2 d t ) p / 2 . Moreover, for $$2\leqslant p<\infty $$ 2 ⩽ p < ∞ one may take $$C_{p,X} = 10 D \sqrt{p},$$ C p , X = 10 D p , where D is the constant in the definition of 2-smoothness for X . The order $$O(\sqrt{p})$$ O ( p ) coincides with that of Burkholder’s inequality and is therefore optimal as $$p\rightarrow \infty $$ p → ∞ . 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 $$g_t\,\mathrm{d} W_t$$ g t d W t is replaced by more general X -valued martingales $$\,\mathrm{d} M_t$$ 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 $$\begin{aligned} \,\mathrm{d} u_t = A(t)u_t\,\mathrm{d} t + g_t\,\mathrm{d} W_t, \quad u_0 = 0, \end{aligned}$$ d u t = A ( t ) u t d t + g t d W t , u 0 = 0 , where the family $$(A(t))_{t\in [0,T]}$$ ( A ( t ) ) t ∈ [ 0 , T ] is assumed to generate a $$C_0$$ C 0 -evolution family $$(S(t,s))_{0\leqslant s\leqslant t\leqslant T}$$ ( S ( t , s ) ) 0 ⩽ s ⩽ t ⩽ 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.