Rolling with Random Slipping and Twisting: A Large Deviation Point of View

TL;DR

This paper analyzes a stochastic model of a rolling system with random slipping and twisting, using large deviation principles to understand its stability and behavior on different manifolds.

Contribution

It establishes large deviation principles for the projection and lift curves of a stochastic rolling model on Riemannian manifolds, including compact and certain noncompact cases.

Findings

Large deviation principles are proved for the projection curves.

Large deviation results are extended to noncompact manifolds in specific cases.

The stability of the stochastic rolling system is characterized probabilistically.

Abstract

We study a rolling model from the perspective of probability. More precisely, we consider a Riemannian manifold rolling against Euclidean space, where the rolling is coupled with random slipping and twisting. The system is modelled by a stochastic differential equation of Stratonovich-type driven by semimartingales, on the orthonormal frame bundle. The stability of the system is examined via large deviations. We prove the large deviation principles for the projection curves on the base manifold and their horizontal lifts respectively, provided that the large deviation holds for the random Euclidean curves as semimartingales. The large deviation results for the case of compact manifolds and two special cases of noncompact manifolds are established.