Remarks on invariance of population distributions for systems with equivariant internal dynamics

TL;DR

This paper discusses how population distributions remain invariant in velocity-jump Markov processes with equivariant internal dynamics, highlighting the role of symmetries in spatial search behaviors.

Contribution

It formalizes the invariance of population distributions in systems with equivariant internal dynamics, connecting symmetry invariance to search performance.

Findings

Population distributions are invariant under certain transformations.

Symmetry invariance explains consistent search behaviors.

Formalization of invariance in velocity-jump processes.

Abstract

For velocity-jump Markov processes with equivariant internal dynamics, we remark that population distributions are invariant. This provides a formalization of the fact that FCD (scale) and other symmetry invariant systems perform identical spatial searches under input transformations.