Quantitative and Subsequence Ergodic Theorems

这一研究建议涉及遍历理论中几乎无处不在的平均收敛问题，以及它们与调和分析和概率论的联系。第一组关于收敛性的问题涉及在事先选定的随机时间序列上对随机过程进行的测量的平均值的平均收敛性。其他问题涉及随机生成的时间，它产生沿大间隙序列采样的结果。第二组问题涉及来自Hardy域成员的子序列的子序列遍历定理。PI在之前的资助期间获得的结果表明，在Hardy域的背景下，对“良好”测量序列的有意义的表征是可能的。事实上，由于在前一个赠款期间取得了重大进展，对良好的测量序列的完整描述即将实现。这项一维平均的工作使我们有信心开始探索可能的高维结果，从而扩展了Stein和Weinger的工作。第三组问题涉及遍历平均的上交叉和相关的振荡行为。这条研究路线是由Bishop、Bourgain、Kalikow、B. Weiss等人发起的。Wierdl和他的合作者发现了遍历理论和鞅之间的基本联系。这一发现——通常表现为能平均和某些鞅之间差异的有界平方函数——允许人们将鞅理论的许多结果，如平方函数、大偏差或跳跃不等式，转化为遍历理论和调和分析结果。提出的部分工作是将对上交叉的研究扩展到其他算子，如高维奇异积分和各种序列域上的平均，并着眼于可能扩展到群行为。这些研究将揭示遍历理论、谐波分析和概率论之间的深刻联系。遍历理论起源于统计力学，即物质的统计描述。后者意味着，例如，不是描述一杯水中每个水分子的行为，而是满足于找到分子的平均速度、能量等。但基本问题随之而来：我们如何测量平均速度或平均能量？显然，测量每个分子的速度，然后取数据的平均值是不可能的。遍历定理说，选择一个分子，测量它每秒的速度就足够了，如果我们进行足够多的测量并取数据的平均值，这个数字基本上就是这杯水中所有分子的平均速度。这个神奇的定理有一个缺点：它要求每一秒都精确地进行测量。但在实践中，测量可能是在1,3,4,6,11…秒，或者更糟，1.1秒，2.4秒，2.9秒，4.3秒……秒，而不是1,2,3，…秒。显然，我们想知道我们是否仍然可以从测量数据中准确地计算出平均速度。提出的研究解决了关于测量的两个基本问题：1)测量的更实际的时间顺序（除了1,2,3…）仍然可以产生平均速度，能量等？2)需要进行多少次测量才能得到平均速度、能量等的有用近似值？注意遍历定理只是说“如果你做了足够的测量，你就会得到关于平均速度，能量等的有用信息”，但它并没有说多少是“足够的”？

This proposal for research concerns questions of almost everywhere and mean convergence in ergodic theory, and their connections with harmonic analysis and probability theory. The first group of questions on convergence which will be addressed concerns the mean convergence of averages of measurements made on a stochastic process at a random sequence of times that is chosen in advance. Other questions concern randomly generated times which yield to sampling along sequences with big gaps. The second group of questions concerns subsequence ergodic theorems for subsequences coming from members of Hardy fields, Results obtained by the PI in the previous grant periods suggest that in the context of Hardy fields, a meaningful characterization of the ``good'' sequences of measurements is possible. Indeed, due to a significant advance in the previous grant period, a complete characterization of good sequences of measurements is at reach. This work for one dimensional averages gives us confidence to start exploring possible higher dimensional results, therefore extending the work of Stein and Weinger. The third group of questions concerns upcrossings and related oscillatory behavior of the ergodic averages. This line of research was initiated by Bishop, Bourgain, Kalikow, B. Weiss and others. Wierdl and his collaborators discovered a fundamental connection between Ergodic Theory and martingales. This discovery---which often manifests itself as a bounded squarefunction of the difference betweenergodic averages and certain martingales---allows one to translate many of the results of Martingale Theory, such as squarefunction, large deviation or jump inequalities, to ergodic theoretical and harmonic analytical results. Part of the proposed work is to extend the investigations on upcrossings to other operators such as higher dimensional singular integrals and averages over various sequences of domains with an eye on possible extensions to group actions. These investigations will reveal deep connections between ergodic theory, harmonic analysis and probability theory. Ergodic theory grew out of statistical mechanics, the statistical description of matter. This latter means, for example, that instead of describing the behavior of each individual water-molecule in a cup ofwater, one is satisfied with finding the average speed, energy etc. of the molecules. But then the fundamental question arises: how can we measure the average speed or energy. It is clearly impossible to measure the speed of each individual molecule and then take the mean of the data. The ergodic theorem says that it is enough to select a single molecule, measure its speed in each second, and if we make enough measurements and take the average of the data, the number will be basically the average speed of all the molecules in the cup of water. This amazing theorem has one drawback: it requires that the measurements are taken exactly at every second. But in practice, the measurements might be made at, say, 1, 3, 4, 6, 11,... seconds or, even worse, at 1.1, 2.4, 2.9, 4.3,... seconds instead of at 1, 2, 3,... seconds. Obviously, we would like to know whether we still can compute accurately the average speed from the measured data. The proposed research addresses two basic questions about measurements: 1) What more practical sequence of times (other than 1, 2, 3,...) for measurements will still yield the average speed, energy, etc.? 2) How many measurements one has to make to get a useful approximation to the average speed, energy, etc.? Note that the ergodic theorem just says "if you make enough measurements, you get useful information about average speed, energy, etc.", but it does not say in any way how many is "enough"?