标题： 随机吸引子和退化扩散的非遍历吸引子 显示英文标题

标题： Random attractors and nonergodic attractors for diffusions with degeneracies

摘要： 我们考虑一个在有界区域上的扩散过程，假设系统在该区域内是不可约的，并且扩散在区域边界上具有不同程度的退化性。 从区域内部出发的典型轨迹的长期统计特性可能由一个不变测度或多个不变测度所支配。 我们描述了各种可能的情况。 在1维和2维情况下，假设边界是双曲的，我们给出了极限行为的完整分类，并回答了涉及多个不变分布的序列平均是否发生的问题。 在所有情况下，我们计算了经验测度的弱极限点集。 用于证明遍历性或非遍历性的击中时间估计基于一种新的Foster-Lyapunov技术版本。 讨论了对非双曲边界和更高维度的扩展，并给出了在可扩展网络中的增长率应用。 显示英文摘要

摘要： We consider a diffusion on a bounded domain, assuming that the system is irreducible inside the domain and that the diffusion has varying degree of degeneracy on the domain's boundary. The long-term statistical properties of typical trajectories started inside the domain may be governed by one invariant measure or more than one invariant measure. We describe various possible scenarios. In dimensions 1 and 2 under boundary hyperbolicity assumptions, we give a complete classification of the limiting behavior and answer the question whether sequential averaging involving more than one invariant distribution occurs. In all cases, we compute the set of weak limit points of empirical measures. Our hitting-time estimates used to prove transience or recurrence are based on a new version of the Foster-Lyapunov technique. Extensions to nonhyperbolic boundaries and higher dimensions are discussed and an application to growth rates in scalable networks is given.