Second order commutator estimates in renormalisation theory for SPDEs with gradient-type noise

TL;DR

This paper addresses the challenge of second order commutator terms arising in renormalisation for SPDEs with gradient-type noise, proving their vanishing under certain conditions, which extends existing theory.

Contribution

It establishes the vanishing of second order commutators in renormalisation for SPDEs with gradient noise on the torus, broadening the scope beyond divergence-free cases.

Findings

Second order commutators vanish for gradient-type noises on  torus.

Extends renormalisation theory to non-divergence-free gradient noise.

Addresses a gap identified in previous work (Punshon-Smith--Smith 2018).

Abstract

An important step in standard renormalisation arguments involve convolution against a standard mollifier. As pointed out in (Punshon-Smith--Smith 2018), this generates second order commutator terms in equations with gradient-type noise. These are commutators similar to commutators in the well-known ``folklore lemma" of Di Perna--Lions (Di Perna--Lions 1989, Lemma II.1), but not covered by standard renormalisation theory. In this note we establish the vanishing of these commutators for gradient-type noises on $\mathbb{T}^d$ not necessarily possessing divergence-free structure.