论文标题
随机微分方程的Lyapunov指数上的下限的规律性方法
A regularity method for lower bounds on the Lyapunov exponent for stochastic differential equations
论文作者
论文摘要
我们提出了一种新方法，用于在随机微分方程（SDE）的顶部Lyapunov指数上获得定量下限。我们的方法结合了（i）（显然是新的）身份，将顶级Lyapunov指数连接到Markov过程的固定密度的类似Fisher信息的功能，该功能与（ii）（ii）（ii）Hörmander的新颖的，定量的版本的hörmander的低纤维化规律性理论在$ l^1 $框架中估计（估计）（nevereate）（neformential frame）。 $ w^{s，1} _ {\ mathrm {loc}} $ sobolev norm。该方法适用于当前现有数学上严格的方法超出范围的广泛系统。作为初始应用，我们证明了一类弱疾病，弱强迫SDE的顶部Lyapunov指数的阳性。在本文中，我们证明，只要将加性随机驾驶应用于任何连续的模式，就包括任何维度的Lorenz 96模型。
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 Fisher information-like functional of the stationary density of the Markov process tracking tangent directions with (ii) a novel, quantitative version of Hörmander's hypoelliptic regularity theory in an $L^1$ framework which estimates this (degenerate) Fisher information from below by a $W^{s,1}_{\mathrm{loc}}$ Sobolev norm. This method is applicable to a wide 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 SDE; 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.
