Stochastic control, entropic interpolation and gradient flows on Wasserstein product spaces

TL;DR

This paper explores the dynamics of entropic interpolation and gradient flows on Wasserstein product spaces, revealing new properties of fluxes and entropy evolution in stochastic control and optimal transport contexts.

Contribution

It introduces a gradient flow on Wasserstein product space with opposite fluxes and derives a novel formula for entropy evolution, extending the understanding of Schrödinger bridge problems.

Findings

Fluxes in the two components are opposite.

The two flows approach each other faster than Fokker-Planck solutions.

Relative entropy can be monotonic in controlled diffusions.

Abstract

Since the early nineties, it has been observed that the Schroedinger bridge problem can be formulated as a stochastic control problem with atypical boundary constraints. This in turn has a fluid dynamic counterpart where the flow of probability densities represents an entropic interpolation between the given initial and final marginals. In the zero noise limit, such entropic interpolation converges in a suitable sense to the displacement interpolation of optimal mass transport (OMT). We consider two absolutely continuous curves in Wasserstein space ${\cal W}_2$ and study the evolution of the relative entropy on ${\cal W}_2\times {\cal W}_2$ on a finite time interval. Thus, this study differs from previous work in OMT theory concerning relative entropy from a fixed (often equilibrium) distribution (density). We derive a gradient flow on Wasserstein product space. We find the remarkable property that fluxes in the two components are opposite. Plugging in the "steepest descent" into the evolution of the relative entropy we get what appears to be a new formula: The two flows approach each other at a faster rate than that of two solutions of the same Fokker-Planck. We then study the evolution of relative entropy in the case of uncontrolled-controlled diffusions. In two special cases of the Schroedinger bridge problem, we show that such relative entropy may be monotonically decreasing or monotonically increasing.