Antoine Gloria, Peter Friz, Nicolas Dirr
Friday, July 6, 2018 - 14:00
HU Berlin
Unter den Linden 6, 10099 Berlin, Lecture room 2094
Antoine Gloria, Sorbonne Unversité Stochastic homogenization: regularity, oscillations, and fluctuations. Abstract: In this colloquial talk I'll give an overview of recent results in stochastic homogenization. Stochastic homogenization is concerned with the analysis of PDEs, say linear elliptic equations in divergence form, with random oscillating coefficients. When the typical correlation-length of the coefficients becomes small, the solution of the PDE can be replaced at first order by the solution of a similar PDE with constant and deterministic coefficients --- this is the qualitative theory of stochastic homogenization. The description of the solution of the original problem by that of the homogenized problem lacks accuracy because the former oscillates (due to the oscillations of the coefficients) and fluctuates (due to the fluctuations of the coefficients) whereas the latter does not. The aim of this talk is precisely to characterize oscillations and fluctuations in this context, for a family of Gaussian coefficient fields (in the full range of decay of the covariance function). Peter Friz, Technische Universität Berlin Rough Paths, Stochastics and PDEs Abstract: I will introduce the basic ideas of Lyons' rough path analysis, taking a numerical analysis perspective. The theory is purely deterministic but allows for many applications to stochastic analysis and PDEs. I will mention large deviations and homogenization as natural application areas of these ideas, making a link to the other talks of this EC Colloquium. Nicolas Dirr, Cardiff Unversity Interacting Particle Systems and Gradient Flows Abstract: One connection between probability and PDE are systems of many randomly interacting particles, which on a much coarser scale are described by a PDEs. In many cases this macroscopic PDE is a gradient flow defined by naturals thermodynamic quantities which are given by the particle system. I will explain this connection and show how it can be used to estimate the macroscopic equation from simulations. This is joint work with Peter Embacher, XIaoguai Li, Celia Reina, Marios Stamatakis, and Johannes Zimmer.
submitted by Ewel (ewel@math.tu-berlin.de, 314 28478)