A priori bounds for the dynamic fractional $\Phi^4$ model on $\mathbb{T}^3$ in the full subcritical regime

TL;DR

This paper establishes a priori bounds for the fractional $\

Contribution

It extends Hairer's regularity structures framework to fractional heat operators, providing new local Schauder estimates and streamlining algebraic methods.

Findings

Proves global existence of solutions for the fractional $\

Establishes invariant measures for the model.

Develops localized multilevel Schauder estimates for fractional heat operators.

Abstract

We show a priori bounds for the dynamic fractional $\Phi^4$ model on $\mathbb{T}^3$ in the full subcritical regime using the framework of Hairer's regularity structures theory. Assuming the model bounds our estimates imply global existence of solutions and existence of an invariant measure. We extend the method developed for the usual heat operator by Chandra, Moinat and Weber [CMW23] to the fractional heat operator, thereby treating a more physically relevant model. A key ingredient in this work is the development of localised multilevel Schauder estimates for the fractional heat operator which is not covered by Hairer's original work. Furthermore, the algebraic arguments from [CMW23] are streamlined significantly.