A regularity method for lower bounds on the Lyapunov exponent for stochastic differential equations

Abstract: In a recent joint work with Alex Blumenthal and Sam Punshon-Smith, we put forward a new method for obtaining quantitative lower bounds on the top Lyapunov exponent of stochastic differential equations (SDEs). Our method combines (i) an (apparently new) identity connecting the top Lyapunov exponent to a degenerate Fisher information-like functional of the stationary density of the Markov process tracking tangent directions with (ii) a quantitative version of Hörmander’s hypoelliptic regularity theory in an L1 framework which estimates this Fisher information from below by a fractional Sobolev norm using the Kolmogorov equation.. As an initial application, we prove the positivity of the top Lyapunov exponent for a class of weakly-dissipative, weakly forced SDE and that this class includes the Lorenz 96 model in any dimension greater than or equal to 7 (as well as finite-dimensional truncations of shell models GOY and SABRA). This is the first mathematically rigorous proof of chaos (in the sense of positive Lyapunov exponents) for stochastically driven Lorenz 96, despite the overwhelming numerical evidence (the deterministic case remains far out of reach).

mathematical physicsanalysis of PDEsdynamical systems

Audience: researchers in the topic

Dynamical systems and PDEs

Series comments: The Zoom link appears on the homepage of the seminar www.mathnet.ru/php/conference.phtml?&eventID=9&confid=1832&option_lang=eng shortly before the talk. You may also subscribe to the mailing list by writing at DSandPDEs at gmail.com