论文标题
关于随机物体产品的注释
A note on products of stochastic objects
论文作者
论文摘要
在最近对具有随机初始数据和奇异随机PDE的偏微分方程（PDE）的研究中，研究各种随机物体的规律性特性至关重要。这些随机对象通常被作为更简单的随机对象的产物。正如Hairer（2014）中指出的那样，通过使用多个随机积分表示，可以使用Jensen的不平等将产品估算到更简单的随机对象上的产品。在本说明中，我们基于Cauchy-Schwarz的不平等（没有任何引用多个随机积分）的简单论点。我们介绍了一个示例，以计算分散性非线性波方程的随机对象的规律性特性，并通过粗糙的随机初始数据证明其局部良好性。
In recent study of partial differential equations (PDEs) with random initial data and singular stochastic PDEs with random forcing, it is essential to study the regularity property of various stochastic objects. These stochastic objects are often given as products of simpler stochastic objects. As pointed out in Hairer(2014), by using a multiple stochastic integral representation, one may use Jensen's inequality to reduce an estimate on the product to those on simpler stochastic objects. In this note, we present a simple argument of the same estimate, based on Cauchy-Schwarz' inequality (without any reference to multiple stochastic integrals). We present an example on computing the regularity property of stochastic objects in the study of the dispersion-generalized nonlinear wave equations, and prove their local well-posedness with rough random initial data.
