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Convergence to the Dynamical $Φ^4_3$ Model under a Vanishing Quintic Perturbation I: A Paracontrolled Approach — arXiv:2609.21552

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.21552
License (as archived): CC0-1.0
Archived: 2026-10-10T02:07:18.118Z
SHA-256 of the archived text: 9ba9f7defdfc48c5c7fd50390a9c22d0edb94023a18de769769cb71e8bb6faae
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How to cite

Convergence to the Dynamical $Φ^4_3$ Model under a Vanishing Quintic Perturbation I: A Paracontrolled Approach — arXiv:2609.21552 — arXiv (en), https://arxiv.org/abs/2609.21552 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_eec2dedae3d4 (Metraipolis archive dc_eec2dedae3d4, sha256 9ba9f7defdfc48c5c7fd50390a9c22d0edb94023a18de769769cb71e8bb6faae)

BibTeX

@misc{dceec2dedae3d4,
  title = {Convergence to the Dynamical $Φ^4_3$ Model under a Vanishing Quintic Perturbation I: A Paracontrolled Approach — arXiv:2609.21552},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.21552},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_eec2dedae3d4; sha256 9ba9f7defdfc48c5c7fd50390a9c22d0edb94023a18de769769cb71e8bb6faae},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.21552. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We study local convergence to the dynamical $\Phi^4_3$ model under a vanishing quintic perturbation. More precisely, on the three-dimensional torus we consider $$\partial_tu_\varepsilon=\Delta u_\varepsilon-\varepsilon^\alpha u_\varepsilon^5+\xi_\varepsilon+C_\varepsilon u_\varepsilon+\widetilde C_\varepsilon u_\varepsilon^3$$ for $\alpha\in(\frac56,1)$, where $\xi_\varepsilon$ is a spatial mollification of space-time white noise. Although the quintic coefficient vanishes, its contractions generate divergent linear and cubic contributions. We identify suitable mass and cubic counterterms that compensate these divergences. Using paracontrolled calculus, we construct the required enhanced stochastic data and prove that their renormalized higher-order components vanish, while the remaining coordinates converge to the enhanced data of the dynamical $\Phi^4_3(\lambda)$ model. We further establish local well-posedness and stability of the associated deterministic solution map. Consequently, for well-prepared initial conditions, $u_\varepsilon$ converges in probability, as a random local solution germ, to the renormalized dynamical $\Phi^4_3(\lambda)$ solution. In particular, the cubic counterterm allows convergence strictly below the threshold $\alpha=1$ arising when only linear renormalization is allowed.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-21 in math.PR).
Categorie arXiv: Probability (math.PR) ; Analysis of PDEs (math.AP).
Pagina del lavoro: https://arxiv.org/abs/2609.21552 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-10T01:47:50.194Z dalla pagina https://arxiv.org/catchup/math.PR/2026-09-21?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).