We present Lyapunov-type conditions for non-strong ergodicity of Markov processes. Some concrete models are discussed, including diffusion processes on Riemannian manifolds and Ornstein–Uhlenbeck processes driven by symmetric $\alpha$-stable processes. In particular, we show that any process of d-dimensional Ornstein–Uhlenbeck type driven by $\alpha$-stable noise is not strongly ergodic for every $\alpha\in (0,2]$.

中文翻译：

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马尔可夫过程非强遍历性的李雅普诺夫型条件

我们提出了马尔可夫过程的非强遍历性的李雅普诺夫型条件。讨论了一些具体模型，包括黎曼流形上的扩散过程和对称驱动的 Ornstein-Uhlenbeck 过程$\阿尔法$- 稳定的过程。特别是，我们表明任何过程d维 Ornstein-Uhlenbeck 类型由$\阿尔法$- 稳定的噪声不是强遍历的$\alpha\in (0,2]$.