A Noether theorem for random locations

TL;DR

This paper introduces a unified framework for analyzing random locations with probabilistic symmetries, establishing a Noether-type theorem that links symmetries to conservation laws in their distributional properties.

Contribution

It develops a novel theoretical framework connecting symmetries in random locations to conservation laws, extending Noether's theorem to probabilistic settings.

Findings

Established a Noether-type theorem for random locations with symmetries

Derived conservation laws for the density functions of random locations

Analyzed boundary behaviors of the distributions

Abstract

We propose a unified framework for random locations exhibiting some probabilistic symmetries such as stationarity, self-similarity, etc. A theorem of Noether's type is proved, which gives rise to a conservation law describing the change of the density function of a random location as the interval of interest changes. We also discuss the boundary and near boundary behavior of the distributions of the random locations.