论文标题
小型扰动可能会改变线性SDE的Lyapunov指数的迹象
Small perturbations may change the sign of Lyapunov exponents for linear SDEs
论文作者
论文摘要
在本文中，我们研究了$ n $维线性随机微分方程（SDE）的存在，因此在呈指数衰减的扰动下，Lyapunov指数的迹象会发生变化。首先，我们表明，所有正面Lyapunov指数的方程式将具有$ n-1 $的独立解决方案，而lyapunov指数为负的方程式。同时，我们证明所有负Lyapunov指数的方程式也将在另一个类似的扰动下具有带有阳性Lyapunov指数的解决方案。最后，我们还表明，出现在方程式不同位置的其他三种扰动将改变Lyapunov指数的迹象。
In this paper, we study the existence of $n$-dimensional linear stochastic differential equations (SDEs) such that the sign of Lyapunov exponents is changed under an exponentially decaying perturbation. First, we show that the equation with all positive Lyapunov exponents will have $n-1$ linearly independent solutions with negative Lyapunov exponents under the perturbation. Meanwhile, we prove that the equation with all negative Lyapunov exponents will also have solutions with positive Lyapunov exponents under another similar perturbation. Finally, we also show that other three kinds of perturbations which appear at different positions of the equation will change the sign of Lyapunov exponents.
