Geometric Langevin equations on submanifolds and applications to the stochastic melt-spinning process of nonwovens and biology

TL;DR

This paper develops geometric Langevin equations on submanifolds, introduces new models with constant velocity, and applies them to fiber lay-down processes in nonwovens and biological systems, connecting geometry with industrial applications.

Contribution

The paper formulates geometric Langevin equations on submanifolds, introduces velocity-constant models, and applies these to fiber lay-down dynamics and particle systems, providing new mathematical tools and industrial models.

Findings

Connected geometric Langevin equations with generalized operators.

Constructed new 3D fiber lay-down models involving spherical Langevin processes.

Linked models to self-propelled particle systems with roosting forces.

Abstract

In this article we develop geometric versions of the classical Langevin equation on regular submanifolds in euclidean space in an easy, natural way and combine them with a bunch of applications. The equations are formulated as Stratonovich stochastic differential equations on manifolds. The first version of the geometric Langevin equation has already been detected before by Leli\`evre, Rousset and Stoltz with a different derivation. We propose an additional extension of the models, the geometric Langevin equations with velocity of constant absolute value. The latters are seemingly new and provide a galaxy of new, beautiful and powerful mathematical models. Up to the authors best knowledge there are not many mathematical papers available dealing with geometric Langevin processes. We connect the first version of the geometric Langevin equation via proving that its generator coincides with the generalized Langevin operator proposed by Soloveitchik, Jorgensen and Kolokoltsov. All our studies are strongly motivated by industrial applications in modeling the fiber lay-down dynamics in the production process of nonwovens. We light up the geometry occuring in these models and show up the connection with the spherical velocity version of the geometric Langevin process. Moreover, as a main point, we construct new smooth industrial relevant three-dimensional fiber lay-down models involving the spherical Langevin process. Finally, relations to a class of self-propelled interacting particle systems with roosting force are presented and further applications of the geometric Langevin equations are given.