Entropy-Maximising Diffusions Satisfy a Parallel Transport Law

Abstract

We show that the principle of maximum entropy, a variational method appearing in statistical inference, statistical physics, and the analysis of stochastic dynamical systems, admits a geometric description from gauge theory. Using the connection on a principal $G$-bundle, the gradient flow maximising entropy is written in terms of constraint functions which interact with the dynamics of the probabilistic degrees of freedom of a diffusion process. This allows us to describe the point of maximum entropy as parallel transport over the state space. In particular, it is proven that the solubility of the stationary Fokker--Planck equation corresponds to the existence of parallel transport in a particular associated vector bundle, extending classic results due to Jordan--Kinderlehrer--Otto and Markowich--Villani. A reinterpretation of splitting results in stochastic dynamical systems is also suggested. Beyond stochastic analysis, we are able to indicate a collection of geometric structures surrounding energy-based inference in statistics.