Title: Spontaneous stochasticity
Abstract: An ordinary differential equation has a unique solution with a given initial condition. This is the fundamental existence and uniqueness theorem, which requires the vector field defining the equation to be smooth (or at least Lipschitz-continuous). It can be expressed in words by saying that a particle moving in a smooth velocity field knows where to go. What if the smoothness condition is violated? Then it may happen that a particle comes to a point from which several trajectories fork out. What is it going to do?
Why is this an interesting question? Because there is strong evidence that velocity fields describing motion of fluid particles in the regime of fully developed turbulence, are so irregular that such forks occur all the time at every point in space. There is an exciting possibility that for some of such fields one can consistently define random dynamics, which intuitively corresponds to the scenario in which the particle chooses from among the available trajectories at random. This idea, which came up in mathematical physics at the end of the last century, is called spontaneous stochasticity. It has many facets and has attracted numerous mathematicians using diverse approaches. This talk will be a nontechnical introduction to spontaneous stochasticity, to some related ideas, and to some mathematical tools that are or may become relevant in their mathematical treatment.