一维对数气体动力学的全局涨落
Global fluctuations for 1D log-gas dynamics
摘要：
在这篇文章中，我们研究了耦合系统的随机微分方程，\begin{equation} dλ_t^i=\frac{1}{\sqrt{N}} dW_t^i - V'(λ_t^i) dt+ \fracβ{2N} \sum_{j\not=i} \frac{dt}{λ^i_t-λ^j_t}, \qquad i=1,\ldots,N, \end{equation}与$β>1$，有时被称为广义戴森布朗运动，描述了一个日志气体的耗散动力学的$N$等电荷与平衡测度对应的$β$ -系综，具有充分规则的凸势$V$在宏观制度的流体力学极限。已知极限$N\to\infty$满足平均场Mac-Kean-Vlasov方程。我们证明了，适当的初始条件下，波动的限制是高斯和满足明确的偏微分方程。这个证明在很大程度上得益于Israelsson \cite{Isr}中所处理的调和势情形。我们的主要论点在于表明，时间演化生成器可以写在上半平面上的运输算子的形式，加上一个有界的非本地运营商解释的一个有符号的跳跃过程。作为保证上述方案收敛性的一个必要的技术论证，我们基于多尺度论证和熵界给出了$\sup_{t\in[0,T]}\max_{i=1,\ldots,N} |λ_t^i|$为大的概率的$N$独立的大偏差型估计。
英文摘要：
We study in this article the hydrodynamic limit in the macroscopic regime of the coupled system of stochastic differential equations, \begin{equation} dλ_t^i=\frac{1}{\sqrt{N}} dW_t^i - V'(λ_t^i) dt+ \fracβ{2N} \sum_{j\not=i} \frac{dt}{λ^i_t-λ^j_t}, \qquad i=1,\ldots,N, \end{equation} with $β>1$, sometimes called generalized Dyson's Brownian motion, describing the dissipative dynamics of a log-gas of $N$ equal charges with equilibrium measure corresponding to a $β$-ensemble, with sufficiently regular convex potential $V$. The limit $N\to\infty$ is known to satisfy a mean-field Mac-Kean-Vlasov equation. We prove that, for suitable initial conditions, fluctuations around the limit are Gaussian and satisfy an explicit PDE. The proof is very much indebted to the harmonic potential case treated in Israelsson \cite{Isr}. Our key argument consists in showing that the time-evolution generator may be written in the form of a transport operator on the upper half-plane, plus a bounded non-local operator interpreted in terms of a signed jump process. As an essential technical argument ensuring the convergence of the above scheme, we give an $N$-independent large-deviation type estimate for the probability that $\sup_{t\in[0,T]}\max_{i=1,\ldots,N} |λ_t^i|$ is large, based on a multi-scale argument and entropic bounds.
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