The theory of rough paths via one-forms and the extension of an argument of Schwartz to rough differential equations

The accessibility problem for geometric rough differential equations

We show how to use geometric arguments to prove that the terminal solution to a rough differential equation driven by a geometric rough path can be obtained by driving the same equation by a piecewise linear path. For this purpose, we combine some results of the seminal work of Sussmann on orbits of vector fields with the rough calculus on manifolds developed by Cass, Litterer and Lyons.

具有粗糙核的非交换奇异积分算子的弱型 $(1,1)$ 有界性

Weak type $(1,1)$ boundedness of noncommutative singular integral operators with rough kernels

本文证明了具有粗糙核的非交换奇异积分算子的弱型 $(1,1)$ 有界性，通过发展核的正交性处理好函数，并利用微局部分解和 $TT^*$ 论证处理坏函数。

Propagation of chaos for mean field rough differential equations

We address propagation of chaos for large systems of rough differential equations associated with random rough differential equations of mean field type $$ dX_t = V(X_t,\mathcal{L}(X_t))dt + F(X_t,\mathcal{L}(X_t))dW_t $$ where $W$ is a random rough path and $\mathcal{L}(X_t)$ is the law of $X_t$. We prove propagation of chaos, and provide also an explicit optimal convergence rate. The analysis is based upon the tools we developed in our companion paper [1] for solving mean field rough differential equations and in particular upon a corresponding version of the Itô-Lyons continuity theorem. Th…

Estimates for singular integrals on homogeneous groups

We consider singular integral operators and maximal singular integral operators with rough kernels on homogeneous groups. We prove certain estimates for the operators that imply $L^p$ boundedness of them by an extrapolation argument under a sharp condition for the kernels. Also, we prove some weighted $L^p$ inequalities for the operators.

粗糙微分方程分裂法的收敛速率

Convergence rate in the splitting-up method for rough differential equations

针对 Hölder 指数在 1/3 到 1/2 之间的粗糙微分方程，提出一种分裂数值格式，并以新论证证明其在 Davie 意义下收敛且给出收敛速率。

光滑粗糙路径、其几何与代数重整化

Smooth rough paths, their geometry and algebraic renormalization

本文引入光滑粗糙路径，利用Maurer-Cartan视角实现Lyons延拓定理的纯代数形式及粗糙路径重整化，并扩展至拟几何与Hopf代数框架。