Rough Burger-like SPDEs

TL;DR

This paper extends the rough path solution theory for Burgers-type stochastic PDEs driven by white noise to a broader regularity range, enabling pathwise solutions in lower regularity regimes.

Contribution

It develops refined estimates and scaling techniques to establish pathwise existence and uniqueness of solutions below the classical rough path threshold.

Findings

Extended solution theory to the full subcritical regime lpha(0, 1/2)

Established new bounds for compositions with smooth functions

Enhanced analytic estimates for rough integrals against heat kernels

Abstract

We study a class of nonlinear Burgers-type stochastic partial differential equations driven by additive space-time white noise in one spatial dimension. Building on the rough path framework initiated by Hairer, which provides a pathwise solution theory under spatial regularity $\alpha \in(\frac{1}{3}, \frac{1}{2})$, we extend this approach to the full subcritical regime $\alpha \in(0, \frac{1}{2})$. Our main contribution is the establishment of pathwise existence and uniqueness of mild (equivalently, weak) solutions when the spatial regularity of the solution lies strictly below the classical rough path threshold. This is achieved through refined estimates for controlled rough paths, including a new upper bound for compositions with smooth functions and a scaling analysis for rough integrals against heat kernels. In particular, we extend and sharpen key analytic estimates originating from Hairer's work, incorporating refined scaling arguments that are effective in the low-regularity regime. As a result, our framework significantly enlarges the class of Burgers-type SPDEs that can be treated pathwise using rough path techniques.