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

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.