Convergence to the Dynamical Φ43 Model under a Vanishing Quintic Perturbation I: A Paracontrolled Approach
Abstract
We study local convergence to the dynamical Φ43 model under a vanishing quintic perturbation. More precisely, on the three-dimensional torus we consider ∂tuε=Δuε−εαuε5+ξε+Cε uε+C̃ε uε3 for α∈(5/6,1), where ξε 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 Φ43(λ) model. We further establish local well-posedness and stability of the associated deterministic solution map. Consequently, for well-prepared initial conditions, uε converges in probability, as a random local solution germ, to the renormalized dynamical Φ43(λ) solution. In particular, the cubic counterterm allows convergence strictly below the threshold α=1 arising when only linear renormalization is allowed.
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BibTeX
@misc{chen2026convergence,
title = {{Convergence to the Dynamical $Φ^4_3$ Model under a Vanishing Quintic Perturbation I: A Paracontrolled Approach}},
author = {Zikai Chen and Seiichiro Kusuoka},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21552},
eprint = {2609.21552},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21552},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Convergence to the Dynamical Φ^4_3 Model under a Vanishing Quintic Perturbation I: A Paracontrolled Approach
AU - Zikai Chen
AU - Seiichiro Kusuoka
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21552
UR - https://arxiv.org/abs/2609.21552
AB - We study local convergence to the dynamical Φ^4_3 model under a vanishing quintic perturbation. More precisely, on the three-dimensional torus we consider ∂_tu_ε=Δu_ε−ε^αu_ε^5+ξ_ε+C_ε u_ε+C̃_ε u_ε^3 for α∈(5/6,1), where ξ_ε 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 Φ^4_3(λ) model. We further establish local well-posedness and stability of the associated deterministic solution map. Consequently, for well-prepared initial conditions, u_ε converges in probability, as a random local solution germ, to the renormalized dynamical Φ^4_3(λ) solution. In particular, the cubic counterterm allows convergence strictly below the threshold α=1 arising when only linear renormalization is allowed.
ER -