Second Order Commutator Estimates in Renormalisation Theory for SPDEs with Gradient-Type Noise

An important step in standard renormalisation arguments involves 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.