从大偏差原理到Wasserstein梯度流：一个新的微观-宏观通道
From a large-deviations principle to the Wasserstein gradient flow: a new micro-macro passage
摘要：
我们一方面研究了多个独立布朗粒子系统与确定性扩散方程之间的联系。对于一个固定的时间步长$h>0$，一个大偏差率泛函$J_h$描述了粒子系统在$t=0$的初始分布在$t=h$的行为。对于扩散方程，在时间离散的熵Wasserstein梯度流的一个单一的步骤，其特征在于功能$K_h$的最小化。通过证明$J_h$和$K_h$在$h$中等于二阶，作为$h\to0$，我们建立了这些系统之间的新联系。这一结果从微观上解释了扩散方程的熵-Wasserstein梯度流公式的起源。同时，这里提出的极限通道通过将其描述为熵梯度流，给出了底层粒子系统的物理自然描述。
英文摘要：
We study the connection between a system of many independent Brownian particles on one hand and the deterministic diffusion equation on the other. For a fixed time step $h>0$, a large-deviations rate functional $J_h$ characterizes the behaviour of the particle system at $t=h$ in terms of the initial distribution at $t=0$. For the diffusion equation, a single step in the time-discretized entropy-Wasserstein gradient flow is characterized by the minimization of a functional $K_h$. We establish a new connection between these systems by proving that $J_h$ and $K_h$ are equal up to second order in $h$ as $h\to0$. This result gives a microscopic explanation of the origin of the entropy-Wasserstein gradient flow formulation of the diffusion equation. Simultaneously, the limit passage presented here gives a physically natural description of the underlying particle system by describing it as an entropic gradient flow.
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