Mathematical Sciences: Universally Good Averaging Sequences in Ergodic Theory

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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). ***