Quantitative and Subsequence Ergodic Theorems

University Of Memphis, Memphis TN

Investigators

Abstract

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 between ergodic 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 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, 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"?