Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Nils Berglund, Yvain Bruned

TL;DR

This work investigates the stochastic PDE ∂_t u − Δ^{ρ/2}u = u^2 + ξ on 𝕋^d in the locally subcritical regime ρ_c < ρ ≤ 2, employing the Bruned–Hairer–Zambotti framework to renormalise the equation via ε-dependent counterterms. The authors provide a precise asymptotic description of the renormalisation constants C_0(ε,ρ) and C_1(ε,ρ) as ε → 0 and ρ ↓ ρ_c, showing a logarithmic scaling away from a superexponentially small ε and a negative-power divergence when ε is superexponentially small. They develop a diagrammatic, BPHZ-based analysis using canonical models, twisted antipodes, forests, Hepp sectors, and detailed bounds on expectations of negative-degree symbols, culminating in explicit regimes determined by thresholds ε_c(ρ) and ε̄_c(ρ). The results illuminate how subcriticality governs renormalisation and offer a concrete illustration of BPHZ renormalisation in a relatively simple near-critical SPDE, with extensions to a broader parameter class. The findings connect near-critical dynamical SPDE renormalisation to static QFT intuition and provide quantitative guidance for the behavior of counterterms in near-critical regimes.

Abstract

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.

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

TL;DR

This work investigates the stochastic PDE ∂_t u − Δ^{ρ/2}u = u^2 + ξ on 𝕋^d in the locally subcritical regime ρ_c < ρ ≤ 2, employing the Bruned–Hairer–Zambotti framework to renormalise the equation via ε-dependent counterterms. The authors provide a precise asymptotic description of the renormalisation constants C_0(ε,ρ) and C_1(ε,ρ) as ε → 0 and ρ ↓ ρ_c, showing a logarithmic scaling away from a superexponentially small ε and a negative-power divergence when ε is superexponentially small. They develop a diagrammatic, BPHZ-based analysis using canonical models, twisted antipodes, forests, Hepp sectors, and detailed bounds on expectations of negative-degree symbols, culminating in explicit regimes determined by thresholds ε_c(ρ) and ε̄_c(ρ). The results illuminate how subcriticality governs renormalisation and offer a concrete illustration of BPHZ renormalisation in a relatively simple near-critical SPDE, with extensions to a broader parameter class. The findings connect near-critical dynamical SPDE renormalisation to static QFT intuition and provide quantitative guidance for the behavior of counterterms in near-critical regimes.

Abstract

We consider stochastic PDEs on the -dimensional torus with fractional Laplacian of parameter , 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 . 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 becomes small and approaches its critical value. In particular, we show that the counterterms behave like a negative power of if is superexponentially small in , and are otherwise of order . This work also serves as an illustration of the general theory of BPHZ renormalisation in a relatively simple situation.

Paper Structure

This paper contains 18 sections, 26 theorems, 246 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model and results
  3. Model space and renormalised equation
  4. Canonical model
  5. Simplifying the twisted antipode
  6. From expectations to Feynman diagrams
  7. Simplification rules for Feynman diagrams
  8. Forests
  9. Zimmermann's forest formula
  10. Hepp sectors and forest intervals
  11. Bounds on $E(\tilde{{\mathcal{A}}}_-^{E}\tau)$
  12. The case ${\mathscr F}_{\mathrm{u}} = \varnothing$
  13. The case ${\mathscr F}_{\mathrm{u}} \neq \varnothing$
  14. Asymptotics
  15. Full binary trees
  16. ...and 3 more sections

Key Result

Theorem 2.1

Assume $\rho < \frac{d}{2}$ and $\rho\in(\rho_{\mathrm{c}},2]$. Then there exist functions $C_i(\varepsilon,\rho)$, $i\in\{0,1\}$, such that for any initial condition $u_0\in {\mathcal{C}}^\eta(\mathbb{T}^d)$ with $\eta>-\frac{\rho}{2}$, the regularised renormalised SPDE eq:renormalised with counter admits a sequence of local solutions $u^\varepsilon$, converging in probability to a limiting proce

Figures (5)

  • Figure 1: Parameter space $(d,\rho)$. Results in this article apply to the locally subcritical regime $\rho_{\mathrm{c}} = \frac{d}{3} < \rho < d$, with $\rho \mathrel\leqsymb 2$.
  • Figure 2: Behaviour of the counterterms as a function of $\rho-\rho_{\mathrm{c}}$ and $\varepsilon$. The small-$\varepsilon$ asymptotics of $C_0$ changes on the blue curve $\varepsilon=\varepsilon_{\mathrm{c}}(\rho)$, while the asymptotics of $C_1$ changes on the green curve $\varepsilon=\bar{\varepsilon}_{\mathrm{c}}(\rho)$.
  • Figure 3: A point configuration $z\in\Lambda^{\mathscr V}$ with its minimal spanning tree (left), and the associated labelled tree $\mathbf{T} = (T(z),\mathbf{n}(z))$ (right). Here ${\mathscr V}=\{v_1,v_2,v_3,v_4,v_5\}$, and node decorations $\mathbf{n}$ are shown in green. For instance, $\mathbf{n}_{v_1\wedge v_2}=2$, so that $z_{v_1}$ and $z_{v_2}$ are at a distance of order $2^{-2}$, while $\mathbf{n}_{v_3\wedge v_5}=1$, so that $z_{v_3}$ and $z_{v_5}$ are at a distance of order $2^{-1}$.
  • Figure 4: A tree $T$ defining a Hepp sector $D_\mathbf{T}$ for the diagram $\mathfrak{K}_{{\mathscr F}_{\mathrm{s}}}\Gamma$ in \ref{['eq:C_gamma12']}. The table shows, for each edge $e$, its image $\sigma(e)=(\sigma(e)_-,\sigma(e)_+)$, the ancestor $e^\uparrow$, the degree of $e$ measured in units of $\frac{d}{3}$ in the limit $\rho\searrow\rho_{\mathrm{c}}$, and the index showing whether the edge has been mollified. Since there can be multiple edges between two given vertices, they have been colour-coded according to their type.
  • Figure 5: A tree $T$ defining a Hepp sector $D_\mathbf{T}$ for the diagram $\hat{{\mathscr C}}_{\gamma_2}\Gamma$, cf. \ref{['eq:Cgamma2']}. The table shows, for each edge, its image $\sigma(e)=(\sigma(e)_-,\sigma(e)_+)$, and the ancestor $e^\uparrow$.

Theorems & Definitions (83)

  • Theorem 2.1: Main result
  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 4
  • Proposition 4.1
  • ...and 73 more