Kolloquium Holzegel/Seis/Weber

Inspired by problems from fluid dynamics, we introduce an approach to obtain lower bounds for Lyapunov exponents of stochastic PDEs. Our proof relies on the introduction of a novel Lyapunov functional for the projective process associated to the equation, based on the study of dynamics of the energy median and on a notion of non-degeneracy of the noise that leads to high-frequency stochastic instability. Joint work with Martin Hairer.

In the complex plane, there is a correspondence between solutions of the Monge?Ampère equation and solutions of a certain differential inclusion associated to SO(2). Under this correspondence, the W^{2,1+epsilon} regularity of solutions to Monge?Ampère is rephrased as a quantitative unique continuation principle for solutions of the differential inclusion. We will sketch a proof of the latter, which relies on quasiconformal maps and the rigidity estimate for SO(2). Based on joint work with G. De Philippis and R. Tione

Large-scale systems of interacting particles are ubiquitous in several relevant applications (in fact, the term particle may stand for actual particles, individuals, animals, parameters in neural networks, etc?). In many circumstances, it may be convenient to describe such particle systems on the continuum scale, i.e., to view their empirical density as the solution of suitable stochastic PDEs. This way of looking at particle systems has given rise to the theory nowadays referred to as Fluctuating Hydrodynamics (FH). In this talk, I plan to (1) give a brief overview of the theory of FH, and (2) discuss high-order numerical discretisations and variance reduction techniques for the Dean-Kawasaki model, a prototype model of FH (based on joint works with Julian Fischer, Jonas Ingmanns, Claudia Raithel)

The macroscopic or mesoscopic behavior of many physical systems interacting with a random environment can be described in terms of non-linear stochastic partial differential equations. Such equations are usually very singular and cannot be solved using standard tools from PDE theory. In the talk, I will present a new technique of solving singular stochastic PDEs based on the renormalization group flow equation. The technique is applicable to a large class of semi-linear parabolic or elliptic SPDEs with fractional Laplacian and covers equations in the whole subcritical regime. A nice feature of the method is that it avoids the algebraic and combinatorial problems arising in different approaches