Analysis, Dynamics and Applications Seminar

For many dynamical systems, e.g. chaotic ones for which individual trajectories evolve in a seemingly random way, it is natural to take a probabilistic perspective and consider the way a randomly chosen initial condition evolves in time. Probability distributions of initial conditions left invariant by the dynamics are called invariant measures. A natural way to probe these invariant measures is to use the Birkhoff ergodic theorem, an analogue of the strong law of large numbers for IID random variables.

Unfortunately, the convergence of this “strong law” for dynamical systems is badly unreliable and slow for many systems of practical interest. Remarkably this is not the case for noisy systems, for which noise has a tendency to regularize statistics. In this (mostly expository) talk, I will review the basics of random dynamical systems, present some compelling case studies, and time permitting will present a theorem giving a quantitative estimate on the rate of convergence of Birkhoff / SLLN for random systems under some mild conditions.

Place:   Hybrid, Math, 402 and Zoom: https://arizona.zoom.us/j/81150211038         Password: “arizona” (all lower case)