Thursday, September 24, 2020
09/24/2020 - 4:15pm
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 (degenerate) Fisher information from below by a fractional, L1-based Sobolev norm using the associated 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 5, provided the additive stochastic driving is applied to any consecutive pair of oscillators. This is the first mathematically rigorous proof of chaos (in the sense of positive Lyapunov exponents) for Lorenz 96 (stochastically driven or otherwise), despite the overwhelming numerical evidence. If time permits, I will also discuss application of the method also to finite dimensional truncations of the classical shell models of hydrodynamic turbulence, GOY and SABRA.