Abstract action spaces and their topological and dynamic properties

Riccarda Rossi; Giuseppe Savaré

doi:10.3934/dcdss.2023208

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2024, Volume 17, Issue 1: 395-420. Doi: 10.3934/dcdss.2023208

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Abstract action spaces and their topological and dynamic properties

Riccarda Rossi^1, , and
Giuseppe Savaré^{1, 2,}

1.
DIMI, Università degli studi di Brescia, Via Branze 38, I–25133 Brescia, Italy
2.
Department of Decision Sciences and BIDSA, Bocconi University, Via Roentgen 1, I–20136 Milan, Italy

^*Corresponding author: Riccarda Rossi
^*Corresponding author: Riccarda Rossi

Dedicated to Pierluigi Colli on the occasion of his 65th birthday

Received: July 2023

Revised: November 2023

Early access: November 2023

Published: January 2024

R.R. acknowledges support from the PRIN project PRIN 2020: "Mathematics for Industry 4.0". G.S. acknowledges support from the PRIN project "PRIN 202244A7YL: Gradient Flows and Non-Smooth Geometric Structures with Applications to Optimization and Machine Learning".

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Abstract

We introduce the concept of action space, a set $ \boldsymbol{X} $ endowed with an action cost $ \mathsf a:(0,+\infty)\times \boldsymbol{X}\times \boldsymbol{X}\to [0,+\infty) $ satisfying suitable axioms, which turn out to provide a 'dynamic' generalization of the classical notion of metric space. Action costs naturally arise as dissipation terms featuring in the Minimizing Movement scheme for gradient flows, which can then be settled in general action spaces.

As in the case of metric spaces, we will show that action costs induce an intrinsic topological and metric structure on $ \boldsymbol{X} $. Moreover, we introduce the related action functional on paths in $ \boldsymbol{X} $, investigate the properties of curves of finite action, and discuss their absolute continuity. Finally, under a condition akin to the approximate mid-point property for metric spaces, we provide a dynamic interpretation of action costs.

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Mathematics Subject Classification: Primary: 35A15; Secondary: 34K30.

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References

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