关于Riemann-Liouville型算子、BMO、Lévy-Itô空间中的梯度估计和逼近
On Riemann-Liouville type operators, BMO, gradient estimates in the Lévy-Itô space, and approximation
摘要：
我们讨论在一个随机框架的Riemann-Liouville型运营商应用于随机过程，真正的插值，有界平均振荡，和随机积分的逼近问题之间的相互作用。我们给出了Lévy-Itô空间上梯度过程的上界和下界，这些上界和下界是由抛物型偏微分方程的Feynman-Kac理论在Wiener空间的特殊情况下得到的.上界由分数阶积分梯度的BMO条件表示，下界由振荡量表示。在一般的Lévy-Itô空间上，我们考虑的是值在希尔伯特空间中的梯度过程，其中正则性依赖于这个希尔伯特空间中的方向。我们讨论了两个应用程序，我们的技术：在Wiener空间的逼近问题的Hölder泛函和Lévy-Itô空间的正交分解的Hölder泛函成一个和随机积分的控制相应的被积。
英文摘要：
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.
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