AI 中文总结
研究非可逆马尔可夫过程的弛豫时间，基于生成器奇异值间隙开发系统方法，能量化弛豫时间，给出精确上下界，还可推导奇异值间隙下界，应用于证明猜想及给出相关过程弛豫时间的界。
AI 中文摘要
我们基于Chatterjee引入的生成器奇异值间隙，开发了一种系统方法来量化非可逆马尔可夫过程的\(L^2\)弛豫时间。奇异值间隙的倒数等同于时间平均转移半群的弛豫时间。此外，两点运动的奇异值间隙为生成器的通常谱间隙提供了下界，其倒数为足够规则初始律下无时间平均的弛豫时间提供了上界。然后我们引入一种基于生成器一阶和二阶坍缩概念的方法来推导具有退化噪声的马尔可夫过程奇异值间隙的下界。它遵循先前一系列工作中发展的hypocoercivity思想，但更简单且更广泛适用。与先前结果不同，它包括一阶坍缩不为零的情况，适用于连续时间马尔可夫链、扩散过程和分段确定性马尔可夫过程。我们的方法为几类示例给出了精确的上下界。首次应用包括证明Diaconis和Miclo关于阿贝尔群上提升随机游走的平方根加速的猜想，以及噪声扰动切换流和非可逆扩散过程弛豫时间的界。
英文摘要
We develop a systematic approach to quantify $L^2$-relaxation times for non-reversible Markov processes based on the singular value gap of the generator introduced by Chatterjee. The inverse of the singular value gap is equivalent to the relaxation time of the time-averaged transition semigroup. We show that, moreover, the singular value gap of the two-point motion also provides a lower bound on the usual spectral gap of the generator, and its inverse provides upper bounds on relaxation times without time averaging for sufficiently regular initial laws. We then introduce a method for deriving lower bounds on singular value gaps for Markov processes with degenerate noise that is based on the concept of a first- and second-order collapse of the generator. It follows ideas from hypocoercivity developed in a previous series of works but is simpler and more broadly applicable. In contrast to previous results, it includes settings with non-vanishing first-order collapse, and thus applies directly to Markov chains (in continuous time), but also to diffusion processes and piecewise-deterministic Markov processes. Our approach yields sharp upper and lower bounds for several classes of examples. First applications include the proof of a conjecture by Diaconis and Miclo on a square-root speed-up for lifted random walks on abelian groups, as well as bounds on relaxation times of switching flows perturbed by noise and of non-reversible diffusion processes.
Comments45 pages
