SPDEs - a rough path perspective

SPDE - 粗略路径视角

• 批准号: 288735431
• 负责人: Professor Dr. Peter Karl Friz
• 金额: --
• 依托单位: Fakultät für Mathematik
• 依托单位国家: 德国
• 项目类别: Research Units
• 财政年份: 2016
• 资助国家: 德国
• 起止时间: 2015-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/288735431?language=en
• 关键词: SPDEs; rough; path; perspective

We develop the mathematical foundations for an existence and uniqueness theory for rough path driven PDEs in the variational framework. In particular, our approach will rely on a suitable extension of the notion of a variational solution to the rough path case, innovative techniques of estimation of rough integrals and the subsequent derivation of basic energy estimates. The methods that will be developed for these purposes will then be applied to study scalar conservation laws and a related class of degenerate parabolic partial differential equations subject to rough perturbation. More precisely, pursuing the so-called kinetic approach we aim at deriving a corresponding notion of kinetic solution and developing a well-posedness theory for these solutions.

我们发展的数学基础的存在性和唯一性理论粗糙路径驱动偏微分方程的变分框架。特别是，我们的方法将依赖于一个适当的扩展的概念变分解决方案的粗糙路径的情况下，创新的技术估计粗糙积分和随后的基本能源估计的推导。将开发用于这些目的的方法，然后将被应用到研究标量守恒律和一类相关的退化抛物型偏微分方程粗糙扰动。更确切地说，追求所谓的动力学方法，我们的目标是推导出相应的动力学解的概念，并发展这些解的适定性理论。