Categorical probability spaces, ergodic decompositions, and transitions to equilibrium
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arXiv
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| Format: | Preprint |
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2023
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| author | Ensarguet, Noé Perrone, Paolo |
| author_facet | Ensarguet, Noé Perrone, Paolo |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_04267 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Categorical probability spaces, ergodic decompositions, and transitions to equilibrium Ensarguet, Noé Perrone, Paolo Probability Logic in Computer Science Category Theory Dynamical Systems 60A05, 18M40, 18M05, 37A50 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. |
| title | Categorical probability spaces, ergodic decompositions, and transitions to equilibrium |
| topic | Probability Logic in Computer Science Category Theory Dynamical Systems 60A05, 18M40, 18M05, 37A50 |
| url | https://arxiv.org/abs/2310.04267 |