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