6 research outputs found
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
Construction of -strong Feller Processes via Dirichlet Forms and Applications to Elliptic Diffusions
We provide a general construction scheme for -strong Feller
processes on locally compact separable metric spaces. Starting from a regular
Dirichlet form and specified regularity assumptions, we construct an associated
semigroup and resolvents of kernels having the -strong Feller
property. They allow us to construct a process which solves the corresponding
martingale problem for all starting points from a known set, namely the set
where the regularity assumptions hold. We apply this result to construct
elliptic diffusions having locally Lipschitz matrix coefficients and singular
drifts on general open sets with absorption at the boundary. In this
application elliptic regularity results imply the desired regularity
assumptions
Hypocoercivity for Kolmogorov backward evolution equations and applications
In this article we extend the modern, powerful and simple abstract Hilbert
space strategy for proving hypocoercivity that has been developed originally by
Dolbeault, Mouhot and Schmeiser. As well-known, hypocoercivity methods imply an
exponential decay to equilibrium with explicit computable rate of convergence.
Our extension is now made for studying the long-time behavior of some strongly
continuous semigroup generated by a (degenerate) Kolmogorov backward operator
L. Additionally, we introduce several domain issues into the framework.
Necessary conditions for proving hypocoercivity need then only to be verified
on some fixed operator core of L. Furthermore, the setting is also suitable for
covering existence and construction problems as required in many applications.
The methods are applicable to various, different, Kolmogorov backward evolution
problems. As a main part, we apply the extended framework to the (degenerate)
spherical velocity Langevin equation. The latter can be seen as some kind of an
analogue to the classical Langevin equation in case spherical velocities are
required. This model is of important industrial relevance and describes the
fiber lay-down in the production process of nonwovens. For the construction of
the strongly continuous contraction semigroup we make use of modern
hypoellipticity tools and pertubation theory