论文标题
2-D随机相互作用的涡旋动力学与单数内核的样品路径大偏差原理
Sample-path large deviation principle for a 2-D stochastic interacting vortex dynamics with singular kernel
论文作者
论文摘要
我们考虑了$ N $颗粒的随机相互作用涡流系统，近似于圆环上2-D Navier-Stokes方程的涡度公式。奇异的相互作用内核由生物 - 萨瓦特法律给出。我们只要求初始状态具有有限的能量，并在涡旋数量进入无穷大时获得经验度量的样品路径大偏差原理。速率函数的特征在于具有有限能量的样本路径上的明确公式，并且随着时间的推移，$ l^2 $规范的有限积分。该证明利用一种对称技术来表示奇异内核，以及具有有限速率函数的样本路径的详细规则分析。关键步骤是证明在对称后的单数项可以通过沿样本路径的$ l^2 $规范的积分来界定。
We consider a stochastic interacting vortex system of $N$ particles, approximating the vorticity formulation of 2-D Navier-Stokes equation on torus. The singular interaction kernel is given by the Biot-Savart law. We only require the initial state to have finite energy, and obtain a sample-path large deviation principle for the empirical measure when the number of vortices goes to infinity. The rate function is characterized by an explicit formula supporting on sample paths with finite energy and finite integral of $L^2$ norms over time. The proof utilizes a symmetrization technique for the representation of singular kernel, together with a detailed regularity analysis of the sample path with finite rate function. The key step is to prove that the singular term after symmetrization can be bounded by the integral of $L^2$ norms along sample paths.
