在消失五次扰动下收敛到动力学 $\Phi^4_3$ 模型 I：一种抛物控制方法

中文摘要 AI 辅助

我们研究在消失五次扰动下局部收敛到动力学 $\Phi^4_3$ 模型的问题。更精确地说，在三维环面上我们考虑 $$\partial_tu_\varepsilon=\Delta u_\varepsilon-\varepsilon^\alpha u_\varepsilon^5+\xi_\varepsilon+C_\varepsilon u_\varepsilon+\widetilde C_\varepsilon u_\varepsilon^3$$ 其中 $\alpha\in(\frac56,1)$，$\xi_\varepsilon$ 是时空白噪声的空间磨光。尽管五次系数消失，其收缩会产生发散线性和三次贡献。我们确定合适的质量和三次反项来补偿这些发散。利用抛物控制微积分，我们构造所需的增强随机数据，并证明其重整化高阶分量消失，而其余坐标收敛到动力学 $\Phi^4_3(\lambda)$ 模型的增强数据。我们进一步建立相关确定性解映射的局部适定性和稳定性。因此，对于准备良好的初始条件，$u_\varepsilon$ 作为随机局部解芽在概率意义下收敛到重整化动力学 $\Phi^4_3(\lambda)$ 解。特别地，三次反项允许收敛严格低于仅允许线性重整化时出现的阈值 $\alpha=1$。

英文摘要

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.

发表机构

• School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）
• Department of Mathematics Sciences, Graduate School of Science, Kyoto University（京都大学理学研究科数学科学系）
• Department of Mathematics and Mathematical Sciences, Graduate School of Science, Kyoto University（京都大学理学研究科数学及数理科学系）

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