Veranstaltungen am Mathematischen Institut

Abstract:
We will describe natural infinite collections of random (Brownian) loops, some of their nice features, and how they are related to some basic ideas and concepts from physics. We will in particular focus on some underlying intriguing random geometric objects in d-dimensional space, and how their construction and behavior drastically differ when d=3, 4 and 5.

About Wendelin Werner:
Wendelin Werner (born 1968) is a German-born French mathematician working on random processes such as self-avoiding random walks, Brownian motion, Schramm-Loewner evolution, and related theories in probability theory and mathematical physics. He is professor at ETH Zurich.

In 2006, he received the Fields Medal for his contribution to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory. He was also honoured with the Rollo Davidson Prize in 1998, the Fermat Prize in 2001, the Grand Prix Jacques Herbrand of the French Academy of Sciences in 2003, the Loève Prize in 2005, the SIAM George Pólya Prize in 2006 and the Heinz Gumin Prize in 2016. He is a member of the French Academy of Sciences, the Academy of Sciences Leopoldina and the Berlin-Brandenburg Academy of Sciences.

It is a classical problem to present a solution of a PDE as the density of the time marginal distributions of a stochastic process. If the PDE is a linear Fokker-Planck equation, then by classical stochastic analysis this is known to be true under very general conditions. For nonlinear Fokker-Planck equations the situation is much more difficult and only known to be true under very restrictive assumptions on the regularity of the (nonlinear) dependence of the coefficients in the Fokker-Planck equations on the solutions. In this talk a new general concept is presented, how to find the desired stochastic process (similarly as in the linear case) through solving a corresponding stochastic differential equation (SDE), whose coefficients, however, depend on the marginal distributions of its solution (DDSDE). The point is that this new general concept does not require strong regularity assumptions on the coefficients (as e.g. fulfilled for McKean-Vlasov type equations) and thus does not rule out a lot of other nonlinear Forker-Planck equations of interest in Physics. As an example it will be shown that it can be applied to the case, where the nonlinear Fokker-Planck equation is a generalized porous media equation on d-dimensional Euclidean space (with d arbitrary), perturbed by a transport term. So its solution is the density of the time marginal distributions of a (tractable) stochastic process solving a corresponding DDSDE. Apart from its conceptual interest this result could lead to new numerical approximations of solutions to nonlinear Fokker-Planck equations through numerically solving the corresponding DDSDE. In the first part of the talk we shall recall the general connection between stochastic differential equations and (both linear and nonlinear) Fokker-Planck equations. Joint work with Viorel Barbu (Romanian Academy of Sciences, Iasi) Reference: arXiv:1801.10510 and SIAM J. Math. Anal. 50 (2018), no. 4, 4246?4260. arXiv:1808.10706