We consider random perturbations of a topologically transitive local diffeomorphism of a Riemannian manifold. We show that if an absolutely continuous ergodic stationary measures is expanding (all Lyapunov exponents positive), then there is a random Gibbs–Markov–Young structure which can be used to lift that measure. We also prove that if the original map admits a finite number of expanding invariant measures then the stationary measures of a sufficiently small stochastic perturbation are expanding.

中文翻译：

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论扩大固定措施的可提升性

我们考虑黎曼流形的拓扑传递局部微分同胚的随机扰动。我们表明，如果一个绝对连续的遍历平稳测度正在扩展（所有 Lyapunov 指数均为正），那么存在一个随机 Gibbs-Markov-Young 结构可以用来提升该测度。我们还证明，如果原始映射允许有限数量的扩展不变测度，那么足够小的随机扰动的平稳测度正在扩展。