Convergence of Single-Timescale Mean-Field Langevin Descent--Ascent for Two-Player Zero-Sum Games

Let $X=Y=\mathbb T^d$, let $f\colon X\times Y\to\mathbb R$ be $C^2$, and let $\beta>0$. For probability measures $\mu,\nu$, define

$$F_\beta(\mu,\nu)=\iint f(x,y)\,d\mu(x)d\nu(y)+\beta^{-1}H(\mu)-\beta^{-1}H(\nu),$$

where $H(\mu)=\int\log(d\mu/dx)\,d\mu$ is negative differential entropy. Let $(\mu_t,\nu_t)$ follow simultaneous Wasserstein gradient descent in $\mu$ and ascent in $\nu$ for $F_\beta$ (the mean-field Langevin descent--ascent flow).

Does $(\mu_t,\nu_t)$ converge weakly as $t\to\infty$ to the unique saddle point $(\mu^\star,\nu^\star)$ for every $f$ and $\beta$? Stronger variants ask for convergence in relative entropy or Nikaido--Isoda error; one may also ask first for local convergence and, once qualitative convergence is settled, for explicit rates.