Extremal flows on Wasserstein space

TL;DR

This paper introduces a geometric framework for calculus of variations on Wasserstein space, unifying various flow models and revealing connections in stochastic mechanics.

Contribution

It develops an intrinsic geometric approach that characterizes key flows as critical points of an action, linking optimal transport, Schrödinger bridges, and fluid dynamics.

Findings

Unified geometric characterization of different flows

Reconciliation between Bohm's and Nelson's stochastic mechanics

Implications for stochastic mechanics and fluid dynamics

Abstract

We develop an intrinsic geometric approach to calculus of variations on Wasserstein space. We show that the flows associated to the Schroedinger bridge with general prior, to Optimal Mass Transport and to the Madelung fluid can all be characterized as annihilating the first variation of a suitable action. We then discuss the implications of this unified framework for stochastic mechanics: It entails, in particular, a sort of fluid-dynamic reconciliation between Bohm's and Nelson's stochastic mechanics.