Mathematics > Operator Algebras
arXiv:2604.12212 (math)
[Submitted on 14 Apr 2026 (v1), last revised 6 Jul 2026 (this version, v2)]
Abstract:We study entropy and optimal transport theory in the free probabilistic setting motivated by the large-$n$ theory of random tuples of matrices. We define a new version of free entropy $\chi_{\operatorname{chron}}^{\mathcal{U}}$, which is concave along geodesics in the corresponding Wasserstein space. Moreover, the heat evolution satisfies the evolution variational inequality, which means that the heat evolution is the Wasserstein gradient flow for entropy in the metric sense. It also has further desirable properties such as a chain rule for iterated conditioning, and an expression in terms of stochastic control problems. This entropy is defined using microstate spaces of matrix approximations with respect to an expanded class of test functions called chronological formulas, which are constructed so as to be closed under taking partial suprema and infima and application of a free heat semigroup. These formulas are part of a novel framework for studying noncommutative filtrations and stochastic processes as metric structures in the sense of continuous model theory.
| Comments: | 94 pages; minor corrections and additions in v2 |
| Subjects: | Operator Algebras (math.OA); Functional Analysis (math.FA); Logic (math.LO); Optimization and Control (math.OC); Probability (math.PR) |
| MSC classes: | Primary: 46L54, 03C66, 49Q22, Secondary: 49J40, 60B20, 60F10, 60G10 |
| Cite as: | arXiv:2604.12212 [math.OA] |
| (or arXiv:2604.12212v2 [math.OA] for this version) | |
| https://doi.org/10.48550/arXiv.2604.12212 arXiv-issued DOI via DataCite |
Submission history
From: David Jekel [view email]
[v1]
Tue, 14 Apr 2026 02:40:20 UTC (93 KB)
[v2]
Mon, 6 Jul 2026 17:06:09 UTC (95 KB)
Current browse context:
math.OA
Bookmark
Bibliographic Tools
Bibliographic and Citation Tools
Bibliographic Explorer Toggle
Code, Data, Media
Code, Data and Media Associated with this Article
Demos
Demos
Related Papers
Recommenders and Search Tools
About arXivLabs
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.