BPHZ renormalisation and vanishing subcriticality asymptotics of the fractional Φ³_d model

Nils Berglund and Yvain Bruned
Stochastics and Partial Differential Equations: Analysis and Computations 13:243-307 (2025)

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.

Mathematical Subject Classification: 60H15, 35R11 (primary), 81T17, 82C28 (secondary).

Keywords and phrases: Stochastic partial differential equations, regularity structures, fractional Laplacian, BPHZ renormalisation, subcriticality boundary.

 

Journal Homepage Published article: 10.1007/s40072-024-00331-2 MR4872109

PDF file (568 Kb) hal-02199627 arXiv/1907.13028