Hostname: page-component-669899f699-tzmfd Total loading time: 0 Render date: 2025-04-28T14:57:38.275Z Has data issue: false hasContentIssue false
  • English
  • Français

Geometric properties of disintegration of measures

Part of: Miscellaneous topics in calculus of variations and optimal control Smooth dynamical systems: general theory Random dynamical systems Optimality conditions

Published online by Cambridge University Press:  11 October 2024

RENATA POSSOBON Open the ORCID record for RENATA POSSOBON [Opens in a new window]  and
CHRISTIAN S. RODRIGUES Open the ORCID record for CHRISTIAN S. RODRIGUES [Opens in a new window]
RENATA POSSOBON
Affiliation:
Institute of Mathematics, Department of Applied Mathematics, Universidade Estadual de Campinas, 13.083-859 Campinas, SP, Brazil (e-mail: [email protected])
CHRISTIAN S. RODRIGUES*
Affiliation:
Institute of Mathematics, Department of Applied Mathematics, Universidade Estadual de Campinas, 13.083-859 Campinas, SP, Brazil (e-mail: [email protected])
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study a connection between disintegration of measures and geometric properties of probability spaces. We prove a disintegration theorem, addressing disintegration from the perspective of an optimal transport problem. We look at the disintegration of transport plans, which are used to define and study disintegration maps. Using these objects, we study the regularity and absolute continuity of disintegration of measures. In particular, we exhibit conditions for which the disintegration map is weakly continuous and one can obtain a path of measures given by this map. We show a rigidity condition for the disintegration of measures to be given into absolutely continuous measures.

Keywords

disintegration theoremprobabilityoptimal transportergodic theorydynamical systems

MSC classification

Primary: 37C40: Smooth ergodic theory, invariant measures 49K45: Problems involving randomness 49N60: Regularity of solutions
Secondary: 37H10: Generation, random and stochastic difference and differential equations 37C05: Smooth mappings and diffeomorphisms
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 45 , Issue 5 , May 2025 , pp. 1619 - 1648
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Ambrosio, L. and Gigli, N.. A User’s Guide to Optimal Transport (Lecture Notes in Mathematics, 2062). Springer-Verlag, Berlin, 2013.Google Scholar
Ambrosio, L., Gigli, N. and Savaré, G.. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Birkhäuser Verlag, Basel, 2005.Google Scholar
Ambrosio, L.. Lecture notes on optimal transport problems. Mathematical Aspects of Evolving Interfaces (Lecture Notes in Mathematics, 1812). Springer, Berlin, 2000.Google Scholar
Ambrosio, L. and Pratelli, A.. Existence and Stability Results in the L1 Theory of Optimal Transportation (Lecture Notes in Mathematics, 1813). Springer, Berlin, 2003.Google Scholar
Butterley, O. and Melbourne, I.. Disintegration of invariant measures for hyperbolic skew products. Israel J. Math. 219 (2017), 171–188.Google Scholar
Chang, J. T. and Pollard, D.. Conditioning as disintegration. Stat. Neerl. 51(3) (1997), 287–317.CrossRefGoogle Scholar
Dellacherie, C.. Ensembles analytiques. Théorèmes de séparation et applications. Séminaire de Probabilités (Lecture Notes in Mathematics, 465). Ed. P. A. Meyer. Springer, Berlin, 1975.Google Scholar
Dellacherie, C. and Meyer, P. A.. Probabilities and Potential (North-Holland Mathematics Studies, 29). North-Holland Publishing Company, Amsterdam, 1978.Google Scholar
Federer, H.. Geometric Measure Theory (Grundlehren der Mathematischen Wissenschaften, 153). Springer, New York, 1969.Google Scholar
Galatolo, S.. Statistical properties of dynamics: introduction to the functional analytic approach. Preprint, 2017, arXiv:1510.02615v2.Google Scholar
Galaz-García, F., Kell, M., Mondino, A. and Sosa, G.. On quotients of spaces with Ricci curvature bounded below. J. Funct. Anal. 275 (2018), 1368–1446.Google Scholar
Galatolo, S. and Lucena, R.. Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps. Discrete Contin. Dyn. Syst. 40 (2020), 1309–1360.Google Scholar
Granieri, L. and Maddalena, F.. Transport problems and disintegration maps. ESAIM Control Optim. Calc. Var. 19(3) (2013), 888–905.Google Scholar
Jost, J.. Riemannian Geometry and Geometric Analysis. Springer-Verlag, Berlin, 2011.CrossRefGoogle Scholar
Oliveira, K. and Viana, M.. Fundamentos da Teoria Ergódica. Sociedade Brasileira de Matemática, Rio de Janeiro, 2014.Google Scholar
Parthasarathy, K. R.. Probability Measures on Metric Spaces. Academic Press, New York, 1967.Google Scholar
Rokhlin, V. A.. On the Fundamental Ideas of Measure Theory (American Mathematical Society Translation, 71). American Mathematical Society, Providence, RI, 1952.Google Scholar
Simmons, D.. Conditional measures and conditional expectation; Rohlin’s disintegration theorem. Discrete Contin. Dyn. Syst. 32(7) (2012), 2565–2582.Google Scholar
Sturm, K. T.. On the geometry of metric measure spaces. Acta Math. 196 (2006), 65–131.Google Scholar
Tjur, T.. A Constructive Definition of Conditional Distributions (Institute of Mathematical Statistics, 13). University of Copenhagen, Copenhagen, 1975.Google Scholar
Varão, R.. Center foliation: absolute continuity, disintegration and rigidity. Ergod. Th. & Dynam. Sys. 36 (2016), 256–275.CrossRefGoogle Scholar
Villani, C.. Topics in Optimal Transportation (Graduate Studies in Mathematics, 58). American Mathematical Society, Providence, RI, 2003.Google Scholar
Villani, C.. Optimal Transport: Old and New (Grundlehren der Mathematischen Wissenschaften, 338). Springer-Verlag, Berlin, 2009.Google Scholar
Von Neumann, J.. Zur Operatorenmethode in der klassischen Mechanik. Ann. of Math. (2) 33 (1932), 587–642.Google Scholar