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

Prof. Jacob Bedrossian, Department of Mathematics, University of Maryland

Chaos is commonly observed in a wide variety of high dimensional systems in physics, however, there are few mathematical tools for obtaining positivity or quantitative estimates on Lyapunov exponents for the vast majority of physical systems. 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) based on the Fisher information of a particular auxiliary Markov process. 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, originally introduced as a toy model for atmospheric dynamics, with any number of unknowns greater than or equal to 7 (the original model has 40). This is the first mathematically rigorous proof of chaos (in the sense of positive Lyapunov exponents) for Lorenz 96, despite the overwhelming numerical evidence of chaos. If time permits, I will also discuss application of the method also to more complicated models such as the finite dimensional truncations of the classical shell models of hydrodynamic turbulence, GOY and SABRA.