Mathematical Sciences: Universally Good Averaging Sequences in Ergodic Theory

[9500577] Wierdl研究者将会——可能会与其他几位研究者合作——研究这个领域最近的一些结果所产生的几乎所有的问题。其中一些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几乎肯定的收敛性。其他问题涉及哈代域成员的子序列遍历定理，沿整数的基，沿具有大间隙的序列。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。但基本的问题出现了：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，如果我们进行足够的测量，并取数据的平均值，这个数字基本上就是一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。但实际上，测量值可能是1.1、2.4、2.9 ....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。拟议的工作从不同的角度（取决于导致测量时间不完美的原因）解决了这个非常困难的问题。＊＊＊

9500577 Wierdl The investigator will work --- possibly in collaboration with several other researchers --- on questions in almost everywhere convergence that come out of some recent results in this area. Some of these questions concern the almost sure convergence of averages of measurements made on a stochastic process at a random sequence of times that is chosen in advance. Other questions concern subsequence ergodic theorems for subsequences coming from members of Hardy fields, along bases for the integers, and along sequences with big gaps. 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 of water, 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, 1.1, 2.4, 2.9 .... 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 work addresses this VERY difficult question from various point of view (depending on WHAT causes the imperfect timing of measurements). ***