论文标题
在随机Camassa-Holm型方程中，具有高阶非线性
On a stochastic Camassa-Holm type equation with higher order nonlinearities
论文作者
论文摘要
本文的主题是在随机扰动下的广义Camassa-Holm方程。我们首先建立本地存在和唯一性结果，以及在Sobolev Spaces $ h^s $的路径解决方案的爆炸标准，$ s> 3/2 $。然后，我们分析噪声如何影响解决方案对初始数据的依赖性。即使噪声具有一些已经知道的正则化效果，但对于对初始数据的依赖性而言，知之甚少。作为一个新概念，我们介绍了退出时间的稳定性概念，并构建了一个示例，表明乘法噪声（从意义上讲）无法提高退出时间的稳定性，并同时提高了对初始数据的依赖性的连续性。最后，我们获得了全球存在定理和估计相关概率。
The subject of this paper is a generalized Camassa-Holm equation under random perturbation. We first establish local existence and uniqueness results as well as blow-up criteria for pathwise solutions in the Sobolev spaces $H^s$ with $s>3/2$. Then we analyze how noise affects the dependence of solutions on initial data. Even though the noise has some already known regularization effects, much less is known concerning the dependence on initial data. As a new concept we introduce the notion of stability of exiting times and construct an example showing that multiplicative noise (in Itô sense) cannot improve the stability of the exiting time, and simultaneously improve the continuity of the dependence on initial data. Finally, we obtain global existence theorems and estimate associated probabilities.
