Caltech/UCLA/USC Joint Analysis Seminar

We review our recent joint work with Alex Blumenthal and Sam Punshon-Smith, which introduced methods for obtaining strictly positive lower bounds on the top Lyapunov exponent of high-dimensional, stochastic differential equations such as the weakly damped Lorenz-96 (L96) model or Galerkin truncations of the 2d Navier-Stokes equations. This hallmark of chaos has long been observed in these models, however, no mathematical proof had previously been made for either deterministic or stochastic forcing. The method is a combination of a new identity connecting the Lyapunov exponents to a Fisher information of the stationary measure of the "projective process" with an L1-based uniform hypoelliptic regularity estimate. We will also discuss some related results, such as dichotomies regarding Lyapunov exponents of general non-dissipative SDEs with applications to chaotic charged particle motion (joint with Chi-Hao Wu) and other applications of uniform hypoelliptic estimates, such as sharp estimates on the spectral gap of Markov semigroups (joint with Kyle Liss).