BPHZ renormalisation and vanishing subcriticality limit of the fractional $Φ^3_d$ model

Nils Berglund, Yvain Bruned

Research output: Working paper

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 $\varepsilon$ becomes small and $\rho$ approaches its critical value. In particular, we show that the counterterms behave like a negative power of $\varepsilon$ if $\varepsilon$ is superexponentially small in $(\rho-d/3)$, and are otherwise of order $\log(\varepsilon^{-1})$. This work also serves as an illustration of the general theory of BPHZ renormalisation in a relatively simple situation.
Original languageEnglish
PublisherArXiv
Publication statusPublished - 30 Jul 2019

Keywords

  • math.PR
  • math-ph
  • math.AP
  • math.MP
  • 60H15, 35R11 (primary), 81T17, 82C28 (secondary)

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