BPHZ renormalisation and vanishing subcriticality limit of the fractional $\Phi^3_d$ 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 $\varepsilon$ 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.
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https://hal.archives-ouvertes.fr/hal-02199627
Contributor : Nils Berglund <>
Submitted on : Wednesday, July 31, 2019 - 8:21:58 AM
Last modification on : Thursday, August 1, 2019 - 2:46:18 AM

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  • HAL Id : hal-02199627, version 1
  • ARXIV : 1907.13028

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Nils Berglund, Yvain Bruned. BPHZ renormalisation and vanishing subcriticality limit of the fractional $\Phi^3_d$ model. 2019. ⟨hal-02199627⟩

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