Related papers: Geometry, mixing properties and hypocoercivity of …
We analyze the large time behavior of a stochastic model for the lay-down of fibers on a conveyor belt in the production process of nonwovens. It is shown, that under weak conditions this degenerate diffusion process is strong mixing,…
This paper concerns models and convergence principles for dealing with stochasticity in a wide range of algorithms arising in nonlinear analysis and optimization in Hilbert spaces. It proposes a flexible geometric framework within which…
The aim of this article is to construct solutions to second order in time stochastic partial differential equations and to show hypocoercivity of the corresponding transition semigroups. More generally, we analyze non-linear…
This note provides an introduction to molecular dynamics, the computational implementation of the theory of statistical physics. The discussion is focused on the properties of Langevin dynamics, a degenerate stochastic differential equation…
We analyze infinite-dimensional non-linear degenerate stochastic differential equations with multiplicative noise. First, essential m-dissipativity of their associated Kolmogorov backward generators on $L^2(\mu^{\Phi})$ defined on smooth…
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…
In this article we investigate hypocoercivity of Langevin-type dynamics in nonlinear smooth geometries. The main result stating exponential decay to an equilibrium state with explicitly computable rate of convergence is rooted in an…
Optimization techniques are at the core of many scientific and engineering disciplines. The steepest descent methods play a foundational role in this area. In this paper we studied a generalized steepest descent method on Riemannian…
In this article we develop a new abstract strategy for proving ergodicity with explicit computable rate of convergence for diffusions associated with a degenerate Kolmogorov operator L. A crucial point is that the evolution operator L may…
A classic approach in dynamical systems is to use particular geometric structures to deduce statistical properties, for example the existence of invariant measures with stochastic-like behaviour such as large deviations or decay of…
A probabilistic approach of computing geometric rate of convergence of stochastic processes is introduced in this paper. The goal is to quantitatively compute both upper and lower bounds of the exponential rate of convergence to the…
We develop an approach to the theory nonholonomic relativistic stochastic processes on curved spaces. The Ito and Stratonovich calculus are formulated for spaces with conventional horizontal (holonomic) and vertical (nonholonomic) splitting…
Stochastic gradient methods are the workhorse (algorithms) of large-scale optimization problems in machine learning, signal processing, and other computational sciences and engineering. This paper studies Markov chain gradient descent, a…
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…
We study the interplay between reversibility, geometry, and the choice of multiplicative noise (in particular It\^{o}, Stratonovich, Klimontovich) in stochastic differential equations (SDEs). Building on a unified geometric framework, we…
Deep generative models learn the data distribution, which is concentrated on a low-dimensional manifold. The geometric analysis of distribution transformation provides a better understanding of data structure and enables a variety of…
We investigate numerical behaviour of a convection diffusion equation with random coefficients by approximating statistical moments of the solution. Stochastic Galerkin approach, turning the original stochastic problem to a system of…
Stochastic geometry provides a powerful framework for modelling complex random structures, with applications in physics, materials science, biology, and other fields. The three-dimensional microstructure of polycrystalline materials is…
We prove existence and uniqueness of stochastic representations for solutions to elliptic and parabolic boundary value and obstacle problems associated with a degenerate Markov diffusion process. In particular, our article focuses on the…
Despite the fact that the theory of mixtures has been part of non-equilibrium thermodynamics and engineering for a long time, it is far from complete. While it is well formulated and tested in the case of mechanical equilibrium (where only…