混合马尔可夫链高维乘积中拟平稳测度的稳定性

中文摘要 AI 辅助

我们研究了由任意数量的独立混合马尔可夫链副本所得到的高维条件动力学。该动力学被条件化以避免一族洞，这些洞的平稳测度随着维数的增长而趋于零，我们关心由此产生的过程是否具有一个接近平稳乘积测度的拟平稳测度。我们的问题受到高维中具有复杂几何形状的洞的条件化所驱动，例如由适当可观测量的大偏差自然产生的集合。我们的主要工具是ANOVA分解，它根据函数对不同坐标子集的依赖性来分离函数。这使我们能够利用动力学的乘积结构，并获得关于维数一致的压缩估计。结合Keller Liverani扰动论证，这些估计产生了拟平稳密度的存在性，并且当洞的大小趋于零时，该密度收敛到平稳密度。在更强的一步混合和正则化假设下，我们在Sobolev范数下获得了更尖锐的收敛性，其依赖于洞的测度的平方根。这一改进依赖于高阶ANOVA分量越来越强的压缩，从而克服了通常随维数增加而失去控制的问题。最后，我们将我们的结果应用于圆上具有光滑、一致正转移密度的加性噪声马尔可夫链。

英文摘要

We study high-dimensional conditioned dynamics obtained from an arbitrary number of independent copies of a mixing Markov chain. The dynamics is conditioned to avoid a family of holes whose stationary measure vanishes as the dimension grows, and we ask whether the resulting process admits a quasi-stationary measure close to the stationary product measure. Our problem is motivated by conditioning on holes with complicated geometry in high dimension, such as sets arising naturally from large deviations of suitable observables. Our main tool is the ANOVA decomposition, which separates functions according to their dependence on different subsets of coordinates. This allows us to exploit the product structure of the dynamics and obtain contraction estimates that are uniform in the dimension. Combined with a Keller Liverani perturbation argument, these estimates yield the existence of quasi-stationary densities converging to the stationary density as the size of the hole vanishes. Under stronger one-step mixing and regularization assumptions, we obtain sharper convergence in a Sobolev norm, with a square-root dependence on the measure of the hole. The improvement relies on the increasingly strong contraction of higher-order ANOVA components, thereby overcoming the usual loss of control with dimension. Finally, we apply our results to additive-noise Markov chains on the circle with smooth, uniformly positive transition densities

发表机构

• King’s College London（伦敦国王学院）
• Imperial College London（帝国理工学院）

机构由 AI 辅助整理，请以论文原文为准。