论文标题
关于Riemann-Liouville类型运营商BMO，Lévy-Itô空间中的梯度估算和近似
On Riemann-Liouville type operators, BMO, gradient estimates in the Lévy-Itô space, and approximation
论文作者
论文摘要
我们在随机框架中讨论了应用于随机过程，实际插值，有界平均振荡的Riemann-Liouville型运算符之间的相互作用，以及随机积分的近似问题。我们为Lévy-Itô空间上的梯度过程提供了上限和下限，这是在Feynman-KAC理论的Wiener Space的特殊情况下出现的，用于抛物线PDES。上限是根据分数综合梯度上的BMO条件来制定的，该梯度是振荡量的下限。在一般的Lévy-Itô空间上，我们关注的是在希尔伯特空间中具有值的梯度过程，其中规律性取决于这个希尔伯特空间的方向。我们讨论了我们技术的两种应用：在维纳空间上，Hölder函数和莱维 - itô空间上的一个近似问题是Hölder函数的正交分解成具有相应集成体的控制的随机积分之和。
We discuss in a stochastic framework the interplay between Riemann-Liouville type operators applied to stochastic processes, real interpolation, bounded mean oscillation, and an approximation problem for stochastic integrals. We provide upper and lower bounds for gradient processes on the Lévy-Itô space, which arise in the special case of the Wiener space from the Feynman-Kac theory for parabolic PDEs. The upper bounds are formulated in terms of BMO-conditions on the fractional integrated gradient, the lower bounds in terms of oscillatory quantities. On the general Lévy-Itô space we are concerned with gradient processes with values in a Hilbert space, where the regularity depends on the direction in this Hilbert space. We discuss two applications of our techniques: on the Wiener space an approximation problem for Hölder functionals and on the Lévy-Itô space an orthogonal decomposition of Hölder functionals into a sum of stochastic integrals with a control of the corresponding integrands.
