Heat semigroup and singular PDEs

TL;DR

This paper introduces a semigroup-based framework for analyzing singular PDEs, extending paracontrolled calculus to unbounded Riemannian manifolds, and demonstrates its effectiveness on the parabolic Anderson model.

Contribution

It develops a heat semigroup approach with a new paraproduct and estimates, enabling the study of singular PDEs on unbounded manifolds under mild conditions.

Findings

Proved well-posedness of the generalized parabolic Anderson model in Hölder spaces.

Extended the machinery to handle more singular settings and Sobolev spaces.

Demonstrated applicability on 2D unbounded Riemannian manifolds with weighted noise.

Abstract

We provide in this work a semigroup approach to the study of singular PDEs, in the line of the paracontrolled approach developed recently by Gubinelli, Imkeller and Perkowski. Starting from a heat semigroup, we develop a functional calculus and introduce a paraproduct based on the semigroup, for which commutator estimates and Schauder estimates are proved, together with their paracontrolled extensions. This machinery allows us to investigate singular PDEs in potentially unbounded Riemannian manifolds under mild geometric conditions. As an illustration, we study the generalized parabolic Anderson model equation and prove, under mild geometric conditions, its well-posed character in Holders spaces, in small time on a potentially unbounded 2-dimensional Riemannian manifold, for an equation driven by a weighted noise, and for all times for the linear parabolic Anderson model equation in 2-dimensional unbounded manifolds. This machinery can be extended to an even more singular setting and deal with Sobolev scales of spaces rather than Holder spaces.