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

• 成果类型: Article
• 署名作者: Bedrossian, Jacob; Blumenthal, Alex; Punshon-Smith, Sam
• 署名单位: University System of Maryland; University of Maryland College Park; University System of Georgia; Georgia Institute of Technology; Institute for Advanced Study - USA
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-021-01069-7
• 发表日期: 2022
• 页码: 429-516
• 关键词: random perturbations; harnack inequality; rotation number; duffing-van; STABILITY; PROOF; bifurcation; Positivity; entropy; systems

摘要

We put forward a new method for obtaining quantitative lower bounds on the top Lyapunov exponent of stochastic differential equations. Our method combines (i) a new identity connecting the top Lyapunov exponent to a Fisher information-like functional of the stationary density of the Markov process tracking tangent directions with (ii) a novel, quantitative version of Hormander's hypoelliptic regularity theory in an L-1 framework which estimates this (degenerate) Fisher information from below by a W-loc(s,1) Sobolev norm. This method is applicable to awide range of systems beyond the reach of currently existing mathematically rigorous methods. As an initial application, we prove the positivity of the top Lyapunov exponent for a class of weakly-dissipative, weakly forced stochastic differential equations; in this paper we prove that this class includes the Lorenz 96 model in any dimension, provided the additive stochastic driving is applied to any consecutive pair of modes.