BPHZ renormalisation and vanishing subcriticality asymptotics of the fractional ${\Phi }_d^3$ model

Abstract

We consider stochastic PDEs on the $d$-dimensional torus with fractional Laplacian of parameter $\rho \in (0,2]$, 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 $\rho >d/3$. 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 $\epsilon$ becomes small and $\rho$ approaches its critical value. In particular, we show that the counterterms behave like a negative power of $\epsilon$ if $\epsilon$ is superexponentially small in $(\rho -d/3)$, and are otherwise of order $\log ({\epsilon }^{-1})$. This work also serves as an illustration of the general theory of BPHZ renormalisation in a relatively simple situation.

arXiv comment52 pages, 4 figures. Extended discussion of the results. Final version, to appear in Stochastics and Partial Differential Equations: Analysis and Computations

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