Probabilistic Representation and Convergence of Gromov-Wasserstein Gradient Flows

Abstract

Wasserstein gradient flows are intimately connected with evolution partial differential equations and diffusion processes. We take the first step in developing such connections for inner product Gromov--Wasserstein (IGW) gradient flows by studying the IGW gradient flow of the relative entropy $\mathsf{H}(\cdot\|\gamma)$ with respect to the standard Gaussian measure $\gamma$. We first show that $\mathsf{H}(\cdot\|\gamma)$ fails to be $\lambda$-convex along generalized or modified generalized IGW geodesics for any $\lambda \in \mathbb{R}$, and therefore falls outside the scope of the existing IGW gradient flow theory from Zhang et al. (2026). We bridge this gap by establishing a suitable \emph{local} convexity estimate that enables the construction of the gradient flow and its extension to the infinite time horizon. We then obtain increasingly explicit representations of the resulting dynamics. Starting from a partial integro-differential equation, we derive a nonlinear Fokker--Planck equation and show that its second-moment dynamics decouple from the law as they satisfy an autonomous matrix ODE. This reduces the IGW dynamics to a linear, time-inhomogeneous Fokker--Planck equation, yielding a probabilistic representation as the time-marginal flow of a linear stochastic differential equation resembling the Ornstein--Uhlenbeck process. Finally, we study its asymptotic behavior by establishing exponential convergence of the flow to $\gamma$ in relative entropy.