We study ordinary differential equations (ODEs) with vector fields given by general Schwartz distributions, and we show that if we perturb such an equation by adding an “infinitely regularizing” path, then it has a unique solution and it induces an infinitely smooth flow of diffeomorphisms. We also introduce a criterion under which the sample paths of a Gaussian process are infinitely regularizing, and we present two processes which satisfy our criterion. The results are based on the path-wise space–time regularity properties of local times, and solutions are constructed using the approach of Catellier–Gubinelli based on nonlinear Young integrals.

中文翻译：

---

受噪声干扰的 ODE 的 C∞− 正则化

我们研究了具有由一般 Schwartz 分布给出的向量场的常微分方程 (ODE)，并且我们表明，如果我们通过添加“无限正则化”路径来扰动这样的方程，那么它就会有一个唯一的解，并且它会产生一个无限平滑的流动微分同胚。我们还引入了一个标准，在该标准下，高斯过程的样本路径是无限正则化的，并且我们提出了两个满足我们标准的过程。结果基于本地时间的路径时空规律性，并使用基于非线性杨积分的 Catellier-Gubinelli 方法构建解决方案。