Distinguished Visitor

Professor Michael Röckner is the immediate past President of the German Mathematical Society (DMV). He received his doctorate from the University of Bielefeld. He became professor in Bonn in 1990 and in Bielefeld in 1994. He is the winner of the Max Planck Research Award 1992 and has been the Dean of the Faculty of Mathematics in Bielefeld.

Professor Röckner works in the area of stochastic analysis. He is one of the most cited authors in the field of stochastic analysis and involved in numerous international activities, which have shaped developments in the field over the last three decades.

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Events

Colloquium 1

Solutions of SPDE as zeros of maps on scaled path spaces

Joint work with Viorel Barbu (Romanian Academy of Sciences, Iasi)

It has been recently shown that the solutions of a large class of stochastic partial differential equations (SPDE) can be obtained as zeros of a properly defined map on a path space equipped with a norm which is “scaled” by the exponential of a function-valued Brownian motion. In the talk this result will be reviewed and connected to current developments about the case where the underlying SPDE is a gradient flow, perturbed by linear multiplicative noise. In this case it follows from the above result and by applying methods from the calculus of variations that the solution minimizes a certain explicit convex functional on the path space. Applications include stochastic porous media equations, stochastic nonlinear parabolic equations (as e.g. the stochastic Cauchy problem for the p-Laplacian) and in the non-gradient case also stochastic nonlinear transport equations.

Colloquium 2

Nonlinear Fokker-Planck equations and distribution dependent SDE

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.

Public Lecture

Finding Order in Disorder: Mathematics is Everywhere

The lecture will present an example of a topic in mathematics which is rooted in elementary school mathematics, materializes later again in high school as well as in university mathematics and finally turns out to be a part of current mathematical research with striking applications, for example in modeling dynamical processes in finance or climatology. Another aim of the lecture is to convince the audience that in order to really understand our environment, both in its phenomenological and quantitative analysis, one has to take into account the influence of randomness.

Coordinator

Dr Tang Wee Kee