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Homotopy probability theory on a Riemannian manifold and the Euler equation

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.