Abstract: 

Optimality is a central concept in mathematical modeling. We are often interested in finding an optimal in a certain sense object. Depending on the application, the​ sense of optimality can be called energy (in physics) or loss (in machine learning). In particular, many classical problems of this type have been successfully solved using methods from the calculus of variations. However, if the optimality criterion becomes random and time-dependent, the standard variational techniques may face significant challenges.

In this lecture I will discuss how randomness interplays with the variational structure. I will first introduce two complementary topics: Gradient Flows and Random Dynamical Systems and highlight a few points of intersection between the two. In the second part I will focus on a new exciting example called Random Quadratic Form (RQF). Combining both the gradient structure and the intrinsic randomness, RQF shows how the variational properties can translate into the probabilistic setting. The last part is based on the joint work with M. Engel  arXiv:2603.06187.