Order Conditions for Sampling the Invariant Measure of Ergodic Stochastic Differential Equations on Manifolds

We derive a new methodology for the construction of high-order integrators for sampling the invariant measure of ergodic stochastic differential equations with dynamics constrained on a manifold. We obtain the order conditions for sampling the invariant measure for a class of Runge–Kutta methods applied to the constrained overdamped Langevin equation. The analysis is valid for arbitrarily high order and relies on an extension of the exotic aromatic Butcher-series formalism. To illustrate the methodology, a method of order two is introduced, and numerical experiments on the sphere, the torus and the special linear group confirm the theoretical findings.

GEOMETRIC LANGEVIN EQUATIONS ON SUBMANIFOLDS AND APPLICATIONS TO THE STOCHASTIC MELT-SPINNING PROCESS OF NONWOVENS AND BIOLOGY

In this paper 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 Lelièvre, Rousset and Stoltz with a different derivation. We propose an additional extension of the models, the geometric Langevin equations with velocity of constant Euclidean norm. 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, Jørgensen or 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 occurring 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 swarming models are presented and further applications of the geometric Langevin equations are given.