论文标题
马尔可夫链在负曲率的流形上的异常复发特性
Anomalous Recurrence Properties of Markov Chains on Manifolds of Negative Curvature
论文作者
论文摘要
我们在具有负曲率的多种流形上提出了一个复发 - 转移分类。我们的分类仅取决于与链的增量相关的几何量，该几何量通过Riemannian指数图定义。我们推断出，马尔可夫链上存在着大量的歧管，它们在每个点都有复发性且平均漂移为零。我们在任意维度的双曲线空间以及随机性不完整的多种歧管上给出了一个明确的例子。我们还证明，与欧几里得的情况相比，这种复发链不能均匀地椭圆形。
We present a recurrence-transience classification for discrete-time Markov chains on manifolds with negative curvature. Our classification depends only on geometric quantities associated to the increments of the chain, defined via the Riemannian exponential map. We deduce that there exist Markov chains on a large class of such manifolds which are both recurrent and have zero average drift at every point. We give an explicit example of such a chain on hyperbolic space of arbitrary dimension, and also on a stochastically incomplete manifold. We also prove that such recurrent chains cannot be uniformly elliptic, in contrast with the Euclidean case.
