Functorial Statistical Physics: Feynman--Kac Formulae and Information Geometries

TL;DR

This paper establishes a functorial framework connecting stochastic differential equations, statistical manifolds, and quantum field theory concepts, extending classical formulas to a broader, time-dependent information geometric setting.

Contribution

It introduces a functorial interpretation of the Feynman--Kac formula and constructs a sheaf-based structure for statistical manifolds, advancing the theoretical foundation of time-dependent statistical physics.

Findings

Feynman--Kac formula as a functor between stochastic and dynamical systems

Statistical manifolds generated by this functor with a sheaf structure

Extension to Chapman--Kolmogorov equation via a generalized maximum entropy principle

Abstract

The main results of this paper comprise proofs of the following two related facts: (i) the Feynman--Kac formula is a functor $F_*$, namely, between a stochastic differential equation and a dynamical system on a statistical manifold, and (ii) a statistical manifold is a sheaf generated by this functor with a canonical gluing condition. Using a particular locality property for $F_*$, recognised from functorial quantum field theory as a `sewing law,' we then extend our results to the Chapman--Kolmogorov equation {\it via} a time-dependent generalisation of the principle of maximum entropy. This yields a partial formalisation of a variational principle which takes us beyond Feynman--Kac measures driven by Wiener laws. Our construction offers a robust glimpse at a deeper theory which we argue re-imagines time-dependent statistical physics and information geometry alike.