Random dynamical systems and stochastic porous media equations with nonlinear noise

The problem of the generation of random dynamical systems by stochastic partial differential equations (SPDE) is one of the open problems in the application of dynamical systems theory to SPDE. Despite its fundamental nature, most results are restricted to ”simple” random perturbations of affine-linear structure. However, as we will see in this course, applications like scaling limits of particle systems with interaction and branching and non-equilibrium statistical mechanics, lead to porous media equations perturbed by nonlinear multiplicative or nonlinear conservative noise. We will first convince ourselves that established methods such as the variational approach to SPDE cannot be applied to these equations, let alone prove the generation of random dynamical systems. Then, based on entropy and kinetic theory we will prove their well-posedness. This will lead to a strong notion of uniqueness, so-called path-by-path uniqueness, based on rough path theory which in turn proves the generation of a corresponding random dynamical system and opens the way to a qualitative analysis of the (stochastic) flow of the solutions.

Date and time info
Monday, 16:15 - 17:45

Keywords
Partial Differential Equations, Applications of PDEs in science

Prerequisites
Basic PDE courses, functional analysis

Audience
MSc students, PhD students, Postdocs

Language
English