Local Algebraic Stability of Wasserstein Gradient Descent-Ascent and Mean Field Langevin Dynamics without Spectral Gap

We study the local stability of Wasserstein gradient descent-ascent (GDA) for entropy regularized two-player zero-sum games on noncompact domains. When one or both equilibrium measures fail to satisfy a Poincaré inequality, the spectral gap arguments underlying local exponential convergence are unavailable. we show that Wasserstein GDA converges algebraically to the mixed Nash equilibrium in chi-squared divergence, provided the initial relative density is a small perturbation in $L^2$ with bounded algebraic moments. The proof has two main ingredients: a cancellation in a negative Sobolev energy estimate, and a bootstrap argument that keeps the moments bounded uniformly in time. For polynomial-tailed equilibria, we construct perturbations supported far out in the tails that survive over diffusive time scales, which gives algebraic lower bounds on the worst-case relaxation rate. We also establish analogous local algebraic stability and lower bound results for mean-field Langevin dynamics, when the equilibrium fails to satisfy a spectral gap.