Abstract
In this paper we discuss the properties of particular set-valued maps in the space of probability measures on a finite-dimensional space that are constructed by mean of a suitable lift of set-valued map in the underlying space. In particular, we are interested to establish under which conditions some good regularity properties of the original set-valued map are inherited by the lifted one. The main motivation for the study is represented by multi-agent systems, i.e., finite-dimensional systems where the number of (microscopic) agents is so large that only macroscopical description are actually available. The macroscopical behaviour is thus expressed by the superposition of the behaviours of the microscopic agents. Using the common description of the state of a multi-agent system by mean of a time-dependent probability measure, expressing the fraction of agents contained in a region at a given time moment, the results of this paper yield regularity results for the macroscopical behaviour of the system.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Ambrosio, L., Gigli, N., Savare, G.: Gradient flows in metric spaces and in the space of probability measures, 2nd edn. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2008)
Aubin, J.-P., Cellina, A.: Differential inclusions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 264. Springer, Cham (1984)
Aubin, J.-P., Frankowska, H.: Set-Valued Analysis, Modern Birkhäuser Classics, Reprint of the 1990 edition [MR1048347]. Birkhäuser Boston Inc., Boston (2009)
Cannarsa, P., Capuani, R.: Existence and uniqueness for mean field games with state constraints. In: Cardaliaguet, P., Porretta, A., Salvarani, F. (eds.) PDE Models for Multi-Agent Phenomena. SIS, vol. 28, pp. 49–71. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-01947-1_3
Cannarsa, P., Capuani, R., Cardaliaguet, P.: \(\bf {C^{1,1}}\)-smoothness of constrained solutions in the calculus of variations with application to mean field games. Math. Eng. 1(1), 174–203 (2018). https://doi.org/10.3934/Mine.2018.1.174
Cannarsa, P., Capuani, R., Cardaliaguet, P.: Mean field games with state constraints: from mild to pointwise solutions of the PDE system. Calculus Variat. Partial Different. Equat. 60(3), 1–33 (2021). https://doi.org/10.1007/s00526-021-01936-4
Lirkov, I., Margenov, S. (eds.): LSSC 2017. LNCS, vol. 10665. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-73441-5
Himmelberg, C.J.: Measurable relations. Fund. Math. 87(1), 53–72 (1975)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 Springer Nature Switzerland AG
About this paper
Cite this paper
Capuani, R., Marigonda, A., Mogentale, M. (2022). Random Lifting of Set-Valued Maps. In: Lirkov, I., Margenov, S. (eds) Large-Scale Scientific Computing. LSSC 2021. Lecture Notes in Computer Science, vol 13127. Springer, Cham. https://doi.org/10.1007/978-3-030-97549-4_34
Download citation
DOI: https://doi.org/10.1007/978-3-030-97549-4_34
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-97548-7
Online ISBN: 978-3-030-97549-4
eBook Packages: Computer ScienceComputer Science (R0)