随机动力学几何遍历性的数值计算
Numerical computations of geometric ergodicity for stochastic dynamics
摘要：
本文介绍一种计算随机过程几何收敛速率的概率方法。目标是定量计算随机过程的不变概率测度的指数收敛速率的上界和下界。应用耦合方法，我们导出了一个不需要离散状态空间的算法。通过这种方式，我们的方法适用于许多高维的例子。我们将此算法应用于随机扰动迭代映射和随机微分方程。我们证明了随机扰动系统的几何遍历率可以在一定程度上揭示未扰动确定性系统的混沌性质。还探讨了各种非线性模型，包括具有退化噪声或高维状态空间的非线性模型。
英文摘要：
A probabilistic approach of computing geometric rate of convergence of stochastic processes is introduced in this paper. The goal is to quantitatively compute both upper and lower bounds of the exponential rate of convergence to the invariant probability measure of a stochastic process. By applying the coupling method, we derive an algorithm which does not need to discretize the state space. In this way, our approach works well for many high-dimensional examples. We apply this algorithm to both random perturbed iterative mappings and stochastic differential equations. We show that the rate of geometric ergodicity of the random perturbed system can, to some extent, reveal the chaotic properties of the unperturbed deterministic one. Various SDE models including those with degenerate noise or high-dimensional state space are also explored.
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