扩散非局部相互作用模型中的维度依赖对称性破缺

中文摘要 AI 辅助

我们研究由随机$n$-单纯形的体积平方生成的扩散非局部相互作用方程。非线性漂移仅通过其均值和协方差矩阵依赖于解，这产生了一个封闭的有限维协方差系统，以及测度值解作为仿射前推后接高斯卷积的显式表示。我们利用这一约化证明了在$\u003cspan_class=\"math\"\u003e$\mathcal P_2(\mathbb R^d)$中任意初始数据的全局适定性。然后我们对稳态进行分类，并证明它们与相关自由能的极小值一致。当$1\leq n\u003cd$时，平衡态是唯一的各向同性高斯分布（平移意义下）。相比之下，在临界维度$n=d$时，能量仅固定协方差矩阵的行列式，并且当$d\geq2$时，出现连续的各向异性高斯平衡态。最后，我们证明了每个解在二次Wasserstein距离下收敛到这些平衡态之一，并获得了显式的指数收敛速率。

英文摘要

We study a diffusive nonlocal interaction equation generated by the squared volumes of random $n$-simplices. The nonlinear drift depends on the solution only through its mean and covariance matrix, which yields a closed finite-dimensional covariance system and an explicit representation of the measure-valued solution as an affine pushforward followed by Gaussian convolution. We use this reduction to prove global well-posedness for arbitrary initial data in $\mathcal P_2(\mathbb R^d)$. We then classify the stationary states and show that they coincide with the minimizers of the associated free energy. When $1\leq n<d$, the equilibrium is a unique isotropic Gaussian up to translation. By contrast, at the critical dimension $n=d$, the energy fixes only the determinant of the covariance matrix, and, when $d\geq2$, a continuum of anisotropic Gaussian equilibria appears. Finally, we prove convergence of every solution to one of these equilibria in quadratic Wasserstein distance and obtain explicit exponential convergence rates.

发表机构

• Sungkyunkwan University（成均馆大学）
• Korea Institute for Advanced Study（韩国高等研究院）
• National Tsing Hua University（国立清华大学）
• Kyungpook National University（庆北国立大学）

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