Malliavin导数和compactness：应用于简并PDE-SDE耦合
The Malliavin derivative and compactness: application to a degenerate PDE-SDE coupling
摘要：
紧性是非线性偏微分方程和系统分析中最通用的工具之一。通常，紧性是通过函数空间之间的嵌入定理来建立的。这样的定理，反过来，依赖于适当的估计，一个功能和它的衍生物。虽然在Wiener-索伯列夫空间上也有类似的基于Malliavin导数和弱Sobolev导数的同时估计的结果，但它似乎还没有被广泛应用于分析具有随机性的高度非线性抛物问题。在目前的工作中，我们应用这一结果，以研究紧凑性，整体解决方案的存在性，并作为一个副产品，一个半离散化计划的一个典型的退化PDE-耦合的收敛性。
英文摘要：
Compactness is one of the most versatile tools in the analysis of nonlinear PDEs and systems. Usually, compactness is established by means of some embedding theorem between functional spaces. Such theorems, in turn, rely on appropriate estimates for a function and its derivatives. While a similar result based on simultaneous estimates for the Malliavin and weak Sobolev derivatives is available for the Wiener-Sobolev spaces, it seems that it has not yet been widely used in the analysis of highly nonlinear parabolic problems with stochasticity. In the present work we apply this result in order to study compactness, existence of global solutions, and, as a by-product, the convergence of a semi-discretisation scheme for a prototypical degenerate PDE-SDE coupling.
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