Physical or biological interpretation of the IGW flow equations

Determine whether the Fokker–Planck equation or stochastic differential equation identified for the inner-product Gromov–Wasserstein gradient flow of relative entropy with respect to the standard Gaussian arises naturally as a model for a physical or biological phenomenon.

Background

The paper derives a nonlinear and then linear Fokker–Planck equation for the inner-product Gromov–Wasserstein gradient flow of relative entropy relative to the standard Gaussian. It also establishes a probabilistic representation of this flow as the time-marginal law of a linear, time-inhomogeneous stochastic differential equation whose coefficients depend on the evolving second-moment matrix.

Although the resulting dynamics resemble the Ornstein–Uhlenbeck process and exhibit exponential convergence to equilibrium, the authors note that they are not generally a deterministic time change of the Ornstein–Uhlenbeck process. The paper leaves unresolved whether either of these newly identified mathematical dynamics has a natural interpretation or application as a model of a physical or biological phenomenon.