Colloquium Mathematics - Prof. Mark Peletier, Technical University Eindhoven

Title: In search of structure: Gradient flows, GENERIC systems, and the role of noise

Abstract:

Many ODEs and PDEs describe real-world phenomena. Often it is
useful to know that these equations have, more "structure" than that
of the bare PDE. For instance, many ODEs and PDEs are Hamiltonian
systems, which provides a wealth of additional information about their
solutions.

Other ODEs and PDEs are _gradient_flows_, and this is the structure
that I will concentrate on. Many evolutionary PDEs are known to be
gradient flows in an appropriate sense, and again this property gives
deep insight and provides many tools for analysis.

Despite the importance of such gradient structures, it is only
relatively recently that we have discovered the reason _why_ many
evolutionary PDEs are gradient flows, in particular why there are so
many gradient flows based on the _Wasserstein_metric_. It turns out
that this is intimately connected to randomness.

In this talk I will discuss gradient flows and their cousin
"GENERIC systems", and show how one can understand how these
deterministic, geometric structures have their roots in randomness.