Fractional Poisson equations and ergodic theorems for fractional coboundaries. (English) Zbl 0988.47009

The main topic of the paper is obtaining speeds of convergence in ergodic theory and probability from assumptions that link the integrable function \(f\) to the transformation under consideration. The paper is based on earlier works C. Kipnis and S. R. Varadhan [Commun. Math. Phys. 104, 1-19 (1986; Zbl 0588.60058)], Y. Derriennic and M. Lin [C. R. Acad. Sci., Paris, Sér. I 323, No. 9, 1053-1057 (1996; Zbl 0869.60019)], A. Brunel [Ann. Inst. Henri Poincaré. n. Ser., Sect. B 9, 327-343 (1974; Zbl 0272.47007)].
For transformation \(T\) of the space \(X\) the elements of \((I-T)X\) are called coboundaries. The characterization \(y\) is a coboundary if and only if \(\sup_n\|\sum_{k=0}^{n-1} T^ky\|<\infty\) is a characterization by the rate of convergence of 1/n in the mean ergodic theorem. For a coboundary \(y\), the Poisson equation \(y=(I-T)x\) can be solved by using the averages of the sequence \(\{\sum_{k=0}^{n-1} T^k y\}\) [see M. Lin and R. Sine, J. Oper. Theory 10, 153-166 (1983; Zbl 0553.47006)].
In the paper, the operator \((I-T)^{\alpha}\) for any \(\alpha\in(0,1)\) is introduced, where \(T\) is a contraction on a Banach space. The elements of the image of \((I-T)^{\alpha}\) are called \(\alpha\)-fractional coboundaries. For \(T\) mean ergodic a characterization of the elements of the image of \((I-T)^{\alpha}\) and a series solution of the fractional Poisson equation \(y=(I-T)^{\alpha}x\) are obtained. The problem for a contraction in a general Banach space is then considered in terms of the norm or the weak topology of the space. Then probability preserving transformations and general Dunford-Schwartz operators are treated from the point of view of the a.e. convergence. Special results for normal contractions in a Hilbert space are given in the last section of the paper.


47A35 Ergodic theory of linear operators
28D05 Measure-preserving transformations
37A30 Ergodic theorems, spectral theory, Markov operators
47A60 Functional calculus for linear operators
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