Transient Acceleration and Cross-Dissipation Interference in Fisher-Regularized Wasserstein Gradient Flows

Abstract

We study transient nonequilibrium dynamics in Fisher-regularized Wasserstein gradient flows and identify a sign-changing cross-dissipation mechanism generated by the coupling between transport dissipation and Fisher-information geometry. Using the Ornstein--Uhlenbeck Fokker--Planck system as an analytically tractable setting, we derive an exact reduced variance dynamics on the Gaussian manifold, \[ \dot{u}=2(1-u)+\frac{\varepsilon}{u}, \] where \(u(t)=\sigma^2(t)\) is the variance and \(\varepsilon>0\) is the Fisher regularization strength. The reduced dynamics reveal distinct transient regimes induced by the interaction between transport relaxation and information-geometric curvature. The associated cross-dissipation term changes sign at the critical scale \(\sigma=1\), separating cooperative acceleration for localized states with \(\sigma<1\) from transient interference at larger variance scales. In the subcritical regime, Fisher curvature accelerates the descent of the baseline free energy; beyond the critical transition, it partially opposes the Ornstein--Uhlenbeck pullback and generates transient overshoot toward a displaced Fisher-regularized equilibrium. We also establish a bounded transient-acceleration-window result, showing that the cooperative acceleration phase has finite duration with an upper bound depending only on the Fisher regularization strength. Finite-difference simulations support the analytical predictions and suggest that qualitatively similar sign-transition behavior may persist beyond Gaussian closure for non-Gaussian initial conditions, including bimodal and Laplace distributions. Overall, the results provide a transient dynamical perspective on Fisher-regularized dissipative systems and show how information-geometric curvature can reorganize intermediate-time Wasserstein relaxation while preserving the globally dissipative structure of the flow.