We study the generalized Dyson Brownian motion (GDBM) of an interacting $N$-particle system with logarithmic Coulomb interaction and general potential $V$. Under reasonable condition on $V$, we prove the existence and uniqueness of strong solution to SDE for GDBM. We then prove that the family of the empirical measures of GDBM is tight on $\mathcal {C}([0,T],\mathscr{P}(\mathbb{R}))$ and all the large $N$ limits satisfy a nonlinear McKean-Vlasov equation. Inspired by previous works due to Biane and Speicher, Carrillo, McCann and Villani, we prove that the McKean-Vlasov equation is indeed the gradient flow of the Voiculescu free entropy on the Wasserstein space of probability measures over $\mathbb{R}$. Using the optimal transportation theory, we prove that if $V"\geq K$ for some constant $K\in \mathbb{R}$, the McKean-Vlasov equation has a unique weak solution. This proves the Law of Large Numbers and the propagation of chaos for the empirical measures of GDBM. Finally, we prove the longtime convergence of the McKean-Vlasov equation for $C^2$-convex potentials $V$.

中文翻译：

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广义戴森布朗运动经验测度过程的大数定律

我们研究具有对数库仑相互作用和一般势 $V$ 相互作用的 $N$-粒子系统的广义戴森布朗运动 (GDBM)。在$V$的合理条件下，我们证明了GDBM的SDE强解的存在唯一性。然后我们证明 GDBM 的经验测量系列在 $\mathcal {C}([0,T],\mathscr{P}(\mathbb{R}))$ 和所有大的 $N$ 限制上是紧密的满足非线性 McKean-Vlasov 方程。受 Biane 和 Speicher、Carrillo、McCann 和 Villani 先前工作的启发，我们证明了 McKean-Vlasov 方程确实是 Voiculescu 自由熵在 $\mathbb{R}$ 上概率测度的 Wasserstein 空间上的梯度流。使用最优运输理论，我们证明如果 $V"\geq K$ 对于某个常数 $K\in \mathbb{R}$，McKean-Vlasov 方程具有唯一的弱解。这证明了GDBM经验测度的大数定律和混沌传播。最后，我们证明了 McKean-Vlasov 方程对于 $C^2$-凸势 $V$ 的长期收敛性。