BPHZ renormalisation and vanishing subcriticality asymptotics of the fractional $Φ^3_d$ model
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866910858730274816 |
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| author | Berglund, Nils Bruned, Yvain |
| author_facet | Berglund, Nils Bruned, Yvain |
| contents | We consider stochastic PDEs on the $d$-dimensional torus with fractional Laplacian of parameter $ρ\in(0,2]$, 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 $ρ> d/3$. 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 $\varepsilon$ becomes small and $ρ$ approaches its critical value. In particular, we show that the counterterms behave like a negative power of $\varepsilon$ if $\varepsilon$ is superexponentially small in $(ρ-d/3)$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1907_13028 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | BPHZ renormalisation and vanishing subcriticality asymptotics of the fractional $Φ^3_d$ model Berglund, Nils Bruned, Yvain Probability Mathematical Physics Analysis of PDEs 60H15, 35R11 (primary), 81T17, 82C28 (secondary) We consider stochastic PDEs on the $d$-dimensional torus with fractional Laplacian of parameter $ρ\in(0,2]$, 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 $ρ> d/3$. 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 $\varepsilon$ becomes small and $ρ$ approaches its critical value. In particular, we show that the counterterms behave like a negative power of $\varepsilon$ if $\varepsilon$ is superexponentially small in $(ρ-d/3)$, 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. |
| title | BPHZ renormalisation and vanishing subcriticality asymptotics of the fractional $Φ^3_d$ model |
| topic | Probability Mathematical Physics Analysis of PDEs 60H15, 35R11 (primary), 81T17, 82C28 (secondary) |
| url | https://arxiv.org/abs/1907.13028 |