Mean-field limit for the complex Euler–Monge–Ampère equation

Let $x_1(t),\ldots,x_N(t)$ be the trajectories of the Hamiltonian flow generated by $E^{(N)}$, and let $\delta_N((x_1(t),\ldots,x_N(t)))$ be their empirical measure. Let $\rho_t$ and $u_{\rho_t}$ satisfy the complex Euler–Monge–Ampère evolution equation on $X$:

$$\frac{\partial\rho_t}{\partial t}=-\nabla\cdot(\rho_tJ\nabla u_{\rho_t}),\qquad \frac{1}{\operatorname{Vol}(L)}(dd^cu_{\rho_t}+\theta)^n=\rho_t\omega_x^n.$$

A solution is smooth when, for every fixed $t>0$, $\rho_t$ and $u_{\rho_t}$ are smooth and $t\mapsto\rho_t(x)$ is differentiable for every fixed $x\in X$; for the initial-value problem, assume $\rho_t\to\rho_0$ in $L^\infty(X)$ as $t\to0$. Complex Euler–Monge–Ampère mean-field conjecture. If at time $t=0$,

$$\lim_{N\rightarrow\infty}\delta_N\left((x_1(t),\ldots,x_N(t))\right)=\rho_0dV,$$

then for every later time $t>0$,

$$\lim_{N\rightarrow\infty}\delta_N\left((x_1(t),\ldots,x_N(t))\right)=\rho_tdV,$$

where $\rho_t$ is a solution of the corresponding initial-value problem for the complex Euler–Monge–Ampère equation. The conjecture asserts propagation of the empirical-measure limit under the many-particle Hamiltonian dynamics and identifies its limiting evolution; the supplied text gives no resolution.