Lecture notes on Malliavin calculus in regularity structures

TL;DR

This paper develops a geometric approach to Malliavin calculus within regularity structures, providing stochastic estimates and a pathwise perspective for PDEs driven by noise, especially in the subcritical regime.

Contribution

It introduces a geometric, model-based framework for Malliavin calculus in regularity structures, linking stochastic estimates to solution manifolds and automating calculus in the subcritical regime.

Findings

Provides a model parameterization of the solution manifold.

Connects Malliavin derivatives to tangent vectors on the manifold.

Automates calculus applicable to the full subcritical regime.

Abstract

Malliavin calculus provides a characterization of the centered model in regularity structures that is stable under removing the small-scale cut-off. In conjunction with a spectral gap inequality, it yields the stochastic estimates of the model. This becomes transparent on the level of a notion of model that parameterizes the solution manifold, and thus is indexed by multi-indices rather than trees, and which allows for a more geometric than combinatorial perspective. In these lecture notes, this is carried out for a PDE with heat operator, a cubic nonlinearity, and driven by additive noise, reminiscent of the stochastic quantization of the Euclidean $\phi^4$ model. More precisely, we informally motivate our notion of the model $(\Pi,\Gamma)$ as charts and transition maps, respectively, of the nonlinear solution manifold. These geometric objects are algebrized in terms of formal power series, and their algebra automorphisms. We will assimilate the directional Malliavin derivative to a tangent vector of the solution manifold. This means that it can be treated as a modelled distribution, thereby connecting stochastic model estimates to pathwise solution theory, with its analytic tools of reconstruction and integration. We unroll an inductive calculus that in an automated way applies to the full subcritical regime.