论文标题
平面中的非线性幼年方程式和随机波方程的噪声途径正规化
Non-linear Young equations in the plane and pathwise regularization by noise for the stochastic wave equation
论文作者
论文摘要
我们根据Catellier和Gubinelli概述的框架精神来研究平面上方程的路线正则化（随机过程。应用，2016年）。为此，我们将非线性年轻方程的概念扩展到二维域，并证明了这种方程的存在和唯一性。然后使用此概念来证明平面上随机方程的噪声正规化。噪声正规化的说明是根据与扰动随机场相关的当地时间的规律性提出的。为此，我们提供了两个量化的例子：一个分数布朗纸和两个单参数分数布朗尼运动的总和。为了进一步说明我们的正则化结果，我们还证明了1D非线性波方程的适合性，该方程具有由分数布朗尼运动给出的嘈杂边界。提供了关于开放问题和进一步调查的讨论。
We study pathwise regularization by noise for equations on the plane in the spirit of the framework outlined by Catellier and Gubinelli (Stochastic Process. Appl., 2016). To this end, we extend the notion of non-linear Young equations to a two dimensional domain and prove existence and uniqueness of such equations. This concept is then used in order to prove regularization by noise for stochastic equations on the plane. The statement of regularization by noise is formulated in terms of the regularity of the local time associated to the perturbing stochastic field. For this, we provide two quantified example: a fractional Brownian sheet and the sum of two one-parameter fractional Brownian motions. As a further illustration of our regularization results, we also prove well-posedness of a 1D non-linear wave equation with a noisy boundary given by fractional Brownian motions. A discussion of open problems and further investigations is provided.
