Noncommutative Regularity Structures

We extend the theory of regularity structures [Hai14] to allow processes belonging to locally m-convex topological algebras. This extension includes processes in the locally C*-algebras of [CHP25] used to localise singular stochastic partial differential equations involving fermions, as well as processes in Banach algebras such as infinite-dimensional semicircular Brownian motion, and more generally the q-Gaussians of [BS91, BKS97, Boż99]. A new challenge we encounter in the q-Gaussian setting with q ∈ (-1,1) are noncommutative renormalisation estimates where we must estimate operators in homogeneous q-Gaussian chaoses with arbitrary operator insertions. We introduce a new Banach algebra norm on q-Gaussian operators that allows us to control such insertions; we believe this construction could be of independent interest.