论文标题
通过层次结构，线性增长和分数噪声，使用$ l^p $相互作用的混乱的熵传播用于平均场扩散
Entropic propagation of chaos for mean field diffusion with $L^p$ interactions via hierarchy, linear growth and fractional noise
论文作者
论文摘要
平均场扩散的混乱结果的新定量传播通过局部和全球熵估计证明。在第一个结果中，我们在圆环上工作，并考虑单数，差异的互动$ k \ in l^p $，$ p> d $。我们证明了$（k^{2}/n^2）$收敛率在$ k $ - 粒子系统的法律及其在每次$ t $时的限制定律之间的相对熵，只要在时间0时保持相同。速率可以在时间均匀地均匀，并在[18]中的结果结合在一起。然后，我们证明了仅认为具有线性生长的相互作用的混乱的定量传播。这将概括为驱动噪声被分数的布朗运动$ b^h $所取代的情况，所有$ h \ in（0,1）$。这些证据来自全球估计值和卧床浓度不平等。在每种情况下，我们在相对熵的相对熵中获得$ O（k/n）$收敛率，但是速率仅在$ [0，t^*] $上有效，$ t^*$带有固定有限常数，具体取决于系统的各种参数。
New quantitative propagation of chaos results for mean field diffusion are proved via local and global entropy estimates. In the first result we work on the torus and consider singular, divergence free interactions $K\in L^p$, $p>d$. We prove a $O(k^{2}/n^2)$ convergence rate in relative entropy between the $k$-marginal laws of the particle system and its limiting law at each time $t$, as long as the same holds at time 0. The proof is based on local estimates via a form of BBGKY hierarchy and exemplifies a method to extend the framework in Lacker [16] to singular interactions. The rate can be made uniform in time combined with the result in [18]. Then we prove quantitative propagation of chaos for interactions that are only assumed to have linear growth. This generalizes to the case where the driving noise is replaced by a fractional Brownian motion $B^H$, for all $H\in(0,1)$. These proofs follow from global estimates and subGaussian concentration inequalities. We obtain $O(k/n)$ convergence rate in relative entropy in each case, yet the rate is only valid on $[0,T^*]$ with $T^*$ a fixed finite constant depending on various parameters of the system.
