Homotopy probability theory on a Riemannian manifold and the Euler equation

TL;DR

This paper introduces homotopy probability theory on Riemannian manifolds, linking it to fluid dynamics by interpreting initial conditions and solutions of the Euler equation as homotopy classes of random variables.

Contribution

It develops a novel framework connecting homotopy probability theory with fluid flow on Riemannian manifolds, providing new insights into the Euler equation.

Findings

Homotopy probability theory can model fluid flow on Riemannian manifolds.

Solutions to the Euler equation correspond to homotopies between homotopy random variables.

The framework offers a new perspective for studying fluid dynamics using algebraic topology.

Abstract

Homotopy probability theory is a version of probability theory in which the vector space of random variables is replaced with a chain complex. A natural example extends ordinary probability theory on a finite volume Riemannian manifold M. In this example, initial conditions for fluid flow on M are identified with collections of homotopy random variables and solutions to the Euler equation are identified with homotopies between collections of homotopy random variables. Several ideas about using homotopy probability theory to study fluid flow are introduced.