Lecture notes on tree-free regularity structures

TL;DR

This paper presents a diagram-free approach to regularity structures, focusing on stochastic estimates of centered models using Malliavin calculus, aimed at understanding a specific class of stochastic PDEs.

Contribution

It introduces a novel diagram-free framework for regularity structures, utilizing Malliavin calculus and spectral gap assumptions for stochastic estimates of centered models.

Findings

Reformulation of centered models via Malliavin derivatives

Development of an annealed Schauder estimate and Liouville principle

Application to a specific parabolic SPDE with renormalization

Abstract

These lecture notes are intended as reader's digest of recent work on a diagram-free approach to the renormalized centered model in Hairer's regularity structures. More precisely, it is about the stochastic estimates of the centered model, based on Malliavin calculus and a spectral gap assumption. We focus on a specific parabolic partial differential equation in quasi-linear form driven by (white) noise. We follow a natural renormalization strategy based on preserving symmetries, and carefully introduce Hairer's notion of a centered model, which provides the coefficients in a formal series expansion of a general solution. We explain how the Malliavin derivative in conjunction with Hairer's re-expansion map allows to reformulate this definition in a way that is stable under removing the small-scale regularization. A few exemplary proofs are provided, both of analytic and of algebraic character. The working horse of the analytic arguments is an ``annealed'' Schauder estimate and related Liouville principle, which is provided. The algebra of formal power series, in variables that play the role of coordinates of the solution manifold, and its algebra morphisms are the key algebraic objects.