Homotopy Probability Theory I

TL;DR

This paper introduces a deformation theoretic framework linking homotopy algebra and probability theory, showing cumulants as morphisms of homotopy algebras, and sets the stage for a homotopy probability theory.

Contribution

It presents the first part of a framework connecting homotopy algebra with probability, emphasizing cumulants as algebra morphisms and enabling derived mathematics in probability theory.

Findings

Cumulants are shown to coincide with morphisms of homotopy algebras.

Framework bridges homotopy algebra and classical/noncommutative probability.

Lays groundwork for developing homotopy probability theory.

Abstract

This is the first of two papers that introduce a deformation theoretic framework to explain and broaden a link between homotopy algebra and probability theory. In this paper, cumulants are proved to coincide with morphisms of homotopy algebras. The sequel paper outlines how the framework presented here can assist in the development of homotopy probability theory, allowing the principles of derived mathematics to participate in classical and noncommutative probability theory.