Lower bounds on the Lyapunov exponents of stochastic differential equations

Abstract

In this article, we review our recently 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 been provided for either deterministic or stochastic forcing.

The method we proposed combines (A) a new identity connecting the Lyapunov exponents to a Fisher information of the stationary measure of the Markov process tracking tangent directions (the so-called “projective process”); and (B) an $L^1$-based hypoelliptic regularity estimate to show that this (degenerate) Fisher information is an upper bound on some fractional regularity. For L96 and GNSE, we then further reduce the lower bound of the top Lyapunov exponent to proving that the projective process satisfies Hörmander’s condition. We review the recent contributions of the first and third authors on the verification of this condition for the 2D Galerkin–Navier–Stokes equations in a rectangular, periodic box of any aspect ratio. Finally, we briefly contrast this work with our earlier work on Lagrangian chaos in the stochastic Navier–Stokes equations. We end the review with a discussion of some open problems.