Higher-Order Paracontrolled Calculus for Geometric Rough Paths

We study the rough differential equation $$\frac{d{u}_t}{dt}=F(u_t){\zeta }_t,u(0)=u_0\in {\mathbb{R}}^d,$$ where $F:{\mathbb{R}}^d\to L({\mathbb{R}}^d,{\mathbb{R}}^d)$ is a sufficiently regular vector field, and the noise $\zeta$ is a distribution in ${\mathcal{C}}^{\alpha -1}(\mathbb{R})$ for $\alpha \in (\frac{1}{4},\frac{1}{3})$. We construct solutions to this equation at lower regularity than previous work of Gubinelli-Imkeller-Perkowski and Prömel-Trabs using ideas from the high-order paracontrolled viewpoint of Bailleul-Bernicot. The key tool in our work is the development of a harmonic-analytic framework for proving a set of iterated commutator estimates which are necessary for the solution theory. We also show the procedure of enhancement when the noise $\zeta =dZ$, where $Z$ is a realization of fractional Brownian motion with Hurst parameter $H\in (\frac{1}{4},\frac{1}{3})$.