论文标题
时间离散方案的随机卷积和路径均匀收敛的最大不平等现象
Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemes
论文作者
论文摘要
我们证明了一个新的Burkholder-Rosenthal类型不平等，用于在2光滑的Banach空间中进行价值的离散时间流程。作为第一个申请，我们证明，如果$（s（t，s））_ {0 \ leq s \ leq t} $是$ c_0 $ - $ 2 $ -2 $ -smooth banach banach space $ x $ x $和$（w_t）_ {t \ in [0，t] $ in [0，T] $ 0 <p <\ infty $存在常数$ c_ {p，x} $，使得所有逐渐可测量的过程$ g：[0，t] \ timesω\ x $ x $ the Process $（\ int_0^t s（t，t，s）g_sdw_s g_sdw_s g_sdw_s） [0，t]} \ big \ | \ int_0^t s（t，s）g_sdw_s \ big \ |^p \ leq c_ { p <\ infty $一个人可能会服用$ c_ {p，x} = 10 d \ sqrt {p}，$ $ d $是$ x $的$ 2 $ -smoothness的定义中的常数。 我们的结果改善并统一了几个现有的最大估计，如果$ x $是希尔伯特空间，则是新的。如果驾驶Martingale $ g_tdw_t $被更通用的$ x $ x $ valued martingales $ dm_t $取代，则获得类似的结果。此外，我们的方法允许随机进化系统，就最大不平等而言，这种设置似乎是全新的。 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 $$ du_t = A(t)u_tdt + g_tdW_t, \quad u_0 = 0,$$ Under spatial smoothness assumptions on the inhomogeneity $g$, contractivity is not needed and explicit decay rates are 获得。在抛物线环境中，这使文献中的几个估计值都更加严格。除了抛物线环境之外，这似乎为时间离散方案提供了第一种系统的系统方法。
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 $C_0$-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 $(Ω,P)$, then for every $0<p<\infty$ there exists a constant $C_{p,X}$ such that for all progressively measurable processes $g: [0,T]\times Ω\to X$ the process $(\int_0^t S(t,s)g_sdW_s)_{t\in [0,T]}$ has a continuous modification and $$E\sup_{t\in [0,T]}\Big\| \int_0^t S(t,s)g_sdW_s \Big\|^p\leq C_{p,X}^p \mathbb{E} \Bigl(\int_0^T \| g_t\|^2_{γ(H,X)}dt\Bigr)^{p/2}.$$ 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 $g_tdW_t$ 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 $$ du_t = A(t)u_tdt + g_tdW_t, \quad u_0 = 0,$$ 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.
