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

Abstract

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 $$\partial_tu_\varepsilon=Δu_\varepsilon-\varepsilon^αu_\varepsilon^5+ξ_\varepsilon+C_\varepsilon u_\varepsilon+\widetilde C_\varepsilon u_\varepsilon^3$$ for $α\in(\frac56,1)$, where $ξ_\varepsilon$ 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 $Φ^4_3(λ)$ model. We further establish local well-posedness and stability of the associated deterministic solution map. Consequently, for well-prepared initial conditions, $u_\varepsilon$ 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.