论文标题
非线性缝纫引理III：稳定性和通用特性
The non-linear sewing lemma III: Stability and generic properties
论文作者
论文摘要
粗糙微分方程（RDE）的溶液可以定义为路径的增量接近相关流的近似值。它们是通过使用非线性缝纫引理的离散方案构建的。在本文中，我们表明，这种解决方案还通过表现出合适的功能来解决固定点问题。然后，融合从一致性和稳定性遵循，这是两个适合我们框架的概念。此外，我们表明，离散近似值的唯一性和收敛性是一种通用属性，这意味着它保留除了一组矢量字段和baire属于baire第一类的起点。最后，我们表明布朗流动几乎是与Lipschitz流相关的RDE的独特解决方案。后来的属性几乎可以确保米尔斯坦计划的收敛性。
Solutions of Rough Differential Equations (RDE) may be defined as paths whose increments are close to an approximation of the associated flow. They are constructed through a discrete scheme using a non-linear sewing lemma. In this article, we show that such solutions also solve a fixed point problem by exhibiting a suitable functional. Convergence then follows from consistency and stability, two notions that are adapted to our framework. In addition, we show that uniqueness and convergence of discrete approximations is a generic property, meaning that it holds excepted for a set of vector fields and starting points which is of Baire first category. At last, we show that Brownian flows are almost surely unique solutions to RDE associated to Lipschitz flows. The later property yields almost sure convergence of Milstein schemes.
