A regularity method for lower bounds on the Lyapunov exponent for stochastic differential equations
A regularity method for lower bounds on the Lyapunov exponent for stochastic differential equations
复制标题
DOI：
10.1007/s00222-021-01069-7
复制
发表时间：
2021-09-13
影响因子：
3.1
通讯作者：
Punshon-Smith, Sam
中科院分区：
文献类型：
--
作者：
Bedrossian, Jacob;Blumenthal, Alex;Punshon-Smith, Sam
我们提出了一种获得随机微分方程顶李雅普诺夫指数的定量下界的新方法。我们的方法结合了（i）一个新的恒等式，将顶部李亚普诺夫指数连接到跟踪切线方向的马尔可夫过程的平稳密度的类似费舍尔信息的函数，与（ii）L-1框架中霍曼德亚椭圆正则理论的新颖定量版本，该框架通过 W-loc(s,1) Sobolev 范数从下面估计这个（简并）费舍尔信息。该方法适用于超出当前现有数学严格方法范围的广泛系统。作为初步应用，我们证明了一类弱耗散、弱受迫随机微分方程的顶部 Lyapunov 指数的正性；在本文中，我们证明该类包括任何维度的 Lorenz 96 模型，前提是将加性随机驱动应用于任何连续的模式对。
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.