Categorical probability spaces, ergodic decompositions, and transitions to equilibrium

TL;DR

This paper develops a categorical framework for probability spaces and Markov kernels, providing new insights into ergodic decompositions and classical results like de Finetti's theorem within this abstract setting.

Contribution

It introduces a categorical approach to probability theory, expressing equilibrium and ergodic results as limits and colimits in a new category of probability spaces.

Findings

Categorical formulation of invariant sigma-algebras as limits and colimits.

Categorical ergodic decomposition theorem for stochastic actions.

Unified perspective on de Finetti, Hewitt-Savage, and Kolmogorov laws.

Abstract

We study a category of probability spaces and measure-preserving Markov kernels up to almost sure equality. This category contains, among its isomorphisms, mod-zero isomorphisms of probability spaces. It also gives an isomorphism between the space of values of a random variable and the sigma-algebra that it generates on the outcome space, reflecting the standard mathematical practice of using the two interchangeably, for example when taking conditional expectations. We show that a number of constructions and results from classical probability theory, mostly involving notions of equilibrium, can be expressed and proven in terms of this category. In particular: - Given a stochastic dynamical system acting on a standard Borel space, we show that the almost surely invariant sigma-algebra can be obtained as a limit and as a colimit; - In the setting above, the almost surely invariant sigma-algebra gives rise, up to isomorphism of our category, to a standard Borel space; - As a corollary, we give a categorical version of the ergodic decomposition theorem for stochastic actions; - As an example, we show how de Finetti's theorem and the Hewitt-Savage and Kolmogorov zero-one laws fit in this limit-colimit picture. This work uses the tools of categorical probability, in particular Markov categories, as well as the theory of dagger categories.