BPHZ renormalisation and vanishing subcriticality asymptotics of the fractional Φd3 model

Abstract

We consider stochastic PDEs on the d-dimensional torus with fractional Laplacian of parameter ρ∈(0,2], quadratic nonlinearity and driven by space-time white noise. These equations are known to be locally subcritical, and thus amenable to the theory of regularity structures, if and only if ρ>d/3. Using a series of recent results by the second named author, A. Chandra, I. Chevyrev, M. Hairer and L. Zambotti, we obtain precise asymptotics on the renormalisation counterterms as the mollification parameter ε becomes small and ρ approaches its critical value. In particular, we show that the counterterms behave like a negative power of ε if ε is superexponentially small in (ρ-d/3), and are otherwise of order log(ε-1). This work also serves as an illustration of the general theory of BPHZ renormalisation in a relatively simple situation.