Alexei A. Mailybaev

Spontaneously stochastic solutions from singularities in differential equations

This talk is about solutions of differential equations that become singular in finite time; the singularity is understood as the lack of Lipschitz continuity. Such singularities are common in applications both for ODEs (e.g., gravitational force at particles collisions) or PDEs (e.g., finite-time blowup in fluid models). The fundamental obstacle is that solutions cannot be continued past the singularity uniquely: typically, there are infinitely many solutions. The conventional way to proceed is to define a regularization limit (such as vanishing viscosity or noise). It turns out that there are structurally stable situations when such a limit is not sensitive to a particular form of the regularization. What is even more surprising is that solutions in this limit may become probabilistic (spontaneously stochastic). I will present some exact results and applications of this phenomenon.