In this paper we develop an existence theory for the Cauchy problem to the stochastic Hunter-Saxton equatio, and prove several properties of the blow-up of its solutions. An important part of the paper is the continuation of solutions to the stochastic equations beyond blow-up (wave-breaking). In the linear noise case, using the method of (stochastic) characteristics, we also study random wave-breaking and stochastic effects unobserved in the deterministic problem. Notably, we derive an explicit law for the random wave-breaking time.

中文翻译：

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带有噪声的 Hunter-Saxton 方程

在本文中，我们为随机 Hunter-Saxton 方程的 Cauchy 问题发展了一个存在理论，并证明了其解爆炸的几个性质。该论文的一个重要部分是在爆炸（破浪）之后对随机方程的解的延续。在线性噪声情况下，使用（随机）特性的方法，我们还研究了确定性问题中未观察到的随机破浪和随机效应。值得注意的是，我们推导出了随机破浪时间的明确定律。