Behaviors of $\phi$-exponential distributions in Wasserstein geometry and an evolution equation

TL;DR

This paper explores the geometric properties of $\,\phi$-exponential distributions within Wasserstein space, revealing their convexity, stability, and new characterizations of Gaussian and q-Gaussian measures.

Contribution

It introduces new geometric insights into $\,\phi$-exponential distributions, including convexity, stability, and characterizations, expanding understanding of their structure in Wasserstein geometry.

Findings

Convexity of $\,\phi$-exponential distributions in Wasserstein space

Stability of these distributions under an evolution equation

New characterizations of Gaussian and q-Gaussian measures

Abstract

A $\phi$-exponential distribution is a generalization of an exponential distribution associated to functions $\phi$ in an appropriate class, and the space of $\phi$-exponential distributions has a dually flat structure. We study features of the space of $\phi$-exponential distributions, such as the convexity in Wasserstein geometry and the stability under an evolution equation. From this study, we provide the new characterizations to the space of Gaussian measures and the space of $q$-Gaussian measures.