Dynamics of geodesic flows with random forcing on Lie groups with left-invariant metrics

TL;DR

This paper investigates how random perturbations affect geodesic flows on Lie groups with left-invariant metrics, focusing on the H"ormander condition and properties of the associated Fokker-Planck equations.

Contribution

It introduces a stochastic framework for geodesic flows on Lie groups and analyzes the H"ormander condition and solution properties of the resulting Fokker-Planck equations.

Findings

H"ormander condition is satisfied under certain stochastic perturbations

Properties of solutions to the Fokker-Planck equations are characterized

Insights into the stochastic dynamics on Lie groups with invariant metrics

Abstract

We consider stochastic perturbations of geodesic flow for left-invariant metrics on finite-dimensional Lie groups and study the H\"ormander condition and some properties of the solutions of the corresponding Fokker-Planck equations.