Abstracts

The Multiplicative Ergodic Theorem (MET) forms the theoretical foundation for many areas of dynamical systems research, notably smooth ergodic theory and the theory of SRB measures for both finite-dimensional systems (ODE and SDE) as well as infinite-dimensional systems (PDE, SPDE, and SDDE). MET is particularly important to study the invariant manifolds, bifurcations, computing the Entropy, and investigating the chaotic behavior.

In this talk, I will talk about the Oseledets splitting (semi-invertible MET) for the singular linear delay equations, and as a result, the existence of invariant manifolds in the non-linear case.

The main tools here are Random dynamical systems and Rough path theory, in particular, our results can also be applied to the singular delay equations driven with FBM with low regularity. This talk is mainly based on the following paper

https://link.springer.com/article/10.1007/s10884-021-09969-1

Joint work with Sebastian Riedel,

In this talk I consider a class of Gaussian random fields on the sphere, obtained as solutions of a certain RPDE. I present a recursive surface finite element method that can be used to generate samples of the fields. I derive convergence rates and analyse the properties of the method.

This talk is based on joint work with Annika Lang and Mihály Kovács.

Motivated by the recent link between stochastic homogenization and rough paths from works by Kelly, Melbourne; Deuschel, Orenshtein and Perkowski, we prove a rough homogenization result for a Brownian particle on a fluctuating Gaussian hypersurface with covariance given by (the ultra-violet cutoff of) the Helfrich energy. The Brownian motion on the surface, whose generator in local coordinates is the Laplace-Beltrami operator, is a simple model for a diffusing particle on a biological surface, which is also known from physics as the overdamped Langevin dynamics on a Helfrich membrane. Considering the membrane moreover fluctuating in time and space (in different speed regimes), Duncan et al. could prove a convergence of the system to the homogenization limit. We extend their results proving the convergence towards a particular lift of the homogenization limit in rough paths topology, for which (in certain regimes) a correction term to the Itô iterated integrals appears. This is ongoing work together with MATH+ AA1-3 project members Ana Djurdjevac, Peter Friz and Nicolas Perkowski.

Consider the stochastic heat equation in dimension one, driven by a space-time white noise. The purpose of this talk is to present a recent result (joint work with David Nualart) on the uniform convergence of the densities of normalized spatial averages of the solution to the density of the standard normal distribution.

In the talk, we investigate models for fibre lay-down processes in industrial production of nonwoven fleeces that are based on the Langevin equations.

We are interested in convergence to equilibrium state given by the abstract Hilbert space hypocoercivity method, since this can be interpreted as a homogeneity result for the nonwoven material. Such results have been obtained for some of the models which we are going to discuss by Grothaus and Stilgenbauer for the case of Euclidean position space.

We show the proper reformulation for abstract Riemannian manifolds as position space such that the hypocoercivity method does apply again and reasonably weak conditions on potential and geometry. We finish with a note on existence of a martingale solution as well as $L^2$-exponential ergodicity of this process.

In this talk, we discuss the asymptotic behaviour of the fluctuations for the empirical measures of interacting particle systems with singular kernels, including the Biot-Savart law. The result applies to the point vortex model approximating the 2D incompressible Navier–Stokes equation and the 2D Euler equation.