论文标题
通过轻度配方的大量相互作用的法律
A Law of Large Numbers for interacting diffusions via a mild formulation
论文作者
论文摘要
考虑一个由独立布朗动作驱动的$ n $弱相互作用粒子的系统。在许多情况下，众所周知，经验措施会收敛到部分微分方程的解决方案，通常称为McKean-Vlasov或Fokker-Planck方程，因为$ n $倾向于无限。我们提出了一种相对较新的方法，通过直接研究经验度量满足每个固定$ n $的随机部分微分方程来证明这种融合。在噪声项的适当控制下，由于系统的有限性，我们能够证明随机扰动为零，表明限制度量是对经典McKean-Vlasov方程的解决方案。与已知结果相反，我们不需要在初始条件上进行任何独立或有限的力矩假设，而是唯一的弱收敛性。在适当的希尔伯特空间中研究了经验度量的演变，其中使用两种不同但互补的技术控制噪声项：粗糙的路径理论和最大程度的自我归一化过程。
Consider a system of $n$ weakly interacting particles driven by independent Brownian motions. In many instances, it is well known that the empirical measure converges to the solution of a partial differential equation, usually called McKean-Vlasov or Fokker-Planck equation, as $n$ tends to infinity. We propose a relatively new approach to show this convergence by directly studying the stochastic partial differential equation that the empirical measure satisfies for each fixed $n$. Under a suitable control on the noise term, which appears due to the finiteness of the system, we are able to prove that the stochastic perturbation goes to zero, showing that the limiting measure is a solution to the classical McKean-Vlasov equation. In contrast with known results, we do not require any independence or finite moment assumption on the initial condition, but the only weak convergence. The evolution of the empirical measure is studied in a suitable class of Hilbert spaces where the noise term is controlled using two distinct but complementary techniques: rough paths theory and maximal inequalities for self-normalized processes.
