Big and noisy: ergodic theory for stochastic and infinite-dimensional dynamical systems

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Not Regularly Scheduled

Real Analysis (at the level of MATH 4317)

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Ergodic theory provides powerful tools for the description of chaotic dynamics. The purpose of this class will be to describe how these tools apply beyond the classical setting of deterministic flows and mappings. After a brief introduction to ergodic theory and some of the classical tools used to describe chaos, I will describe how these tools can be extended to treat certain classes of (a) stochastic flows, generated by stochastic differential equations; and (b)infinite-dimensional dynamics, with an eye to semiflows generated by dissipative parabolic partial differential equations (e.g., reaction-diffusion equations, some pattern-formation systems, and some fluids models such as Navier-Stokes). Preliminaries will be kept minimal as possible, but some measure theory will be required -- to be reviewed early on in the class.