平均场颗粒介质方程的长时大偏差渐近

中文摘要 AI 辅助

我们研究了颗粒介质扩散的长时平均场大偏差。在耗散性假设下，对于任意$T_N\to\infty$，$N$粒子经验测度的占据测度在$\mathcal{P}(\mathcal{P}(\mathbb{R}^d))$上满足大偏差原理，速度为$NT_N$，速率函数为$\mathcal{I}(\Gamma)=\int_{\mathcal{P}(\mathbb{R}^d)}\mathfrak{i}(\mu)\Gamma(d\mu)$。这里$\mathfrak{i}(\mu)$是自由能耗散的四分之一：$\mathfrak{i}(\mu)=\frac14\left\\|\nabla\frac{\delta\mathcal{F}}{\delta\mu}(\mu)\right\\|_{L^2(\mu)}^2$，其中$\mathcal{F}$是模型的自由能。局部成本$\mathfrak{i}(\mu)$可以解释为常数路径在$\mu$处的每单位时间Dawson--Gärtner作用，而$\mathcal{I}$根据$\Gamma$对这些成本进行平均。上界的主要困难在于经验测度是原子的，因此它们的自由能是无限的。我们通过沿着受控颗粒介质扩散的经验测度流分析正则化的自由能泛函来解决这一问题。下界使用有限支撑的占据测度来近似，这些测度的原子具有高斯尾部的光滑密度。对于二次势，我们证明了与异常大的整体时空方差相关的变分问题表现出狄拉克极小值（在高斯轮廓处）与非狄拉克极小值（支撑在不同中心的高斯轮廓上）之间的转变。

英文摘要

We study long-time mean-field large deviations for granular-media diffusions. Under a dissipativity assumption, for arbitrary $T_N\to\infty$, the occupation measure of the $N$-particle empirical measure satisfies a large deviation principle on $\mathcal{P}(\mathcal{P}(\mathbb{R}^d))$ with speed $NT_N$ and rate function $\mathcal{I}(Γ)=\int_{\mathcal{P}(\mathbb{R}^d)}\mathfrak{i}(μ)Γ(dμ)$. Here $\mathfrak{i}(μ)$ is one quarter of the free-energy dissipation: $\mathfrak{i}(μ)=\frac14\left\|\nabla\frac{δ\mathcal{F}}{δμ}(μ)\right\|_{L^2(μ)}^2$, where $\mathcal{F}$ is the free energy of the model. The local cost $\mathfrak{i}(μ)$ can be interpreted as the per-unit-time Dawson--Gärtner action of the constant path at $μ$, and $\mathcal{I}$ averages these costs according to $Γ$. The main difficulty in the upper bound is that empirical measures are atomic, so their free energy is infinite. We address this by analyzing a regularized free-energy functional along the empirical-measure flow of controlled granular-media diffusions. The lower bound uses approximation by finitely supported occupation measures whose atoms have smooth densities with Gaussian tails. For quadratic potentials, we show that the variational problem associated with an atypically large overall space-time variance exhibits a transition between a Dirac minimizer at a Gaussian profile and non-Dirac minimizers supported on Gaussian profiles with different centers.

发表机构

• University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）

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57 pages