×

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.

MSC:

47A35 Ergodic theory of linear operators
28D05 Measure-preserving transformations
37A30 Ergodic theorems, spectral theory, Markov operators
47A60 Functional calculus for linear operators
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] [A] I. Assani,Wiener-Wintner dynamical systems, preprint, 1998. · Zbl 0957.37002
[2] [ANi] I. Assani and K. Nicolaou,Properties of Wiener-Wintner dynamical systems, preprint, 1999. · Zbl 0994.37004
[3] [BoI] A. N. Borodin and I. A. Ibragimov,Limit Theorems for Functionals of Random Walks, Proceedings of the Steklov Institute of Mathematics195 (1994); English translation: American Mathematical Society, Providence, RI, 1995. · Zbl 0840.60002
[4] [BDenDe] M. Broise, Y. Déniel and Y. Derriennic,Réarrangement, inégalités maximales et théorèmes ergodiques fractionnaires, Annales de l’Institut Fourier39 (1989), 689–714. · Zbl 0673.60055
[5] [Br] A. Brunel,Théorème ergodique ponctuel pour un semigroupe finiment engendré de contractions de L 1, Annales de l’Institut Henri Poincaré. Probabilités et Statistiques9 (1973), 327–343.
[6] [BuW] P. Butzer and U. Westphal,The mean ergodic theorem and saturation, Indiana University Mathematics Journal20 (1971), 1163–1174. · Zbl 0217.45202
[7] [C] J. Campbell,Spectral analysis of the ergodic Hilbert transform, Indiana University Mathematics Journal35 (1986), 379–390. · Zbl 0575.47023
[8] [ÇLi] D. Çömez and M. Lin,Mean ergodicity of L 1 contractions and pointwise ergodic theorems, inAlmost Everywhere Convergence II (A Bellow and R. Jones, eds.), Academic Press, Boston, 1991, pp. 113–126. · Zbl 0759.47004
[9] [DRo] A. Del Junco and J. Rosenblatt,Counterexamples in ergodic theory and number theory, Mathematische Annalen245 (1979), 185–197. · Zbl 0408.28015
[10] [Den] Y. Déniel,On the a.s. Cesàro-{\(\alpha\)} convergence for stationary or orthogonal random variables, Journal of Theoretical Probability2 (1989), 475–485. · Zbl 0684.60014
[11] [DeLi] Y. Derriennic and M. Lin,Sur le théorème limite central de Kipnis et Varadhan pour les chaînes réversibles ou normales, Comptes Rendus de l’Académie des Sciences, Paris, Série I323 (1996), 1053–1057. · Zbl 0869.60019
[12] [DuS] N. Dunford and J. Schwartz,Linear Operators I, Interscience, New York, 1958.
[13] [F] S. R. Foguel,The Ergodic Theory of Markov Processes, Van Nostrand, New York, 1969.
[14] [G-1] V. Gaposhkin,Convergence of a series related to stationary processes, Izvestia Matematika39 (1975), 1366–1392 (in Russian).
[15] [G-2] V. Gaposhkin,On the dependence of the convergence rate in the SLLN for stationary processes on the rate of decay of the correlation function, Theory of Probability and its Applications26 (1981), 706–720. · Zbl 0488.60040
[16] [G-3] V. Gaposhkin,Spectral criteria for existence of generalized ergodic transforms, Theory of Probability and its Applications41 (1996), 247–264. · Zbl 0881.60038
[17] [GoLif-1] M. Gordin and B. Lifŝic (Lifshits),A central limit theorem for Markov processes, Soviet Mathematics Doklady19 (1978), 392–394.
[18] [GoLif-2] M. Gordin and B. Lifŝic (Lifshits),A remark about a Markov process with normal transition operator, Third Vilnius Conference on Probability Theory and Mathematical Statistics, Vol. 1, Akademiya Nauk Litovsk, Vilnius, 1981, pp. 147–148 (in Russian).
[19] [H] P. R. Halmos,A non-homogeneous ergodic theorem, Transactions of the American Mathematical Society66 (1949), 284–288. · Zbl 0036.20502
[20] [Ho] S. Horowitz,Pointwise convergence of the iterates of a Harris-recurrent Markov operator, Israel Journal of Mathematics33 (1979), 177–180. · Zbl 0435.60068
[21] [K] A. G. Kachurovskii,The rate of convergence in ergodic theorems, Russian Mathematical Surveys51 (1996), 653–703. · Zbl 0880.60024
[22] [KiV] C. Kipnis and S. R. Varadhan,Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions, Communications in Mathematical Physics104 (1986), 1–19. · Zbl 0588.60058
[23] [Kr] U. Krengel,Ergodic Theorems, de Gruyter, Berlin, 1985.
[24] [L] M. Lacey,On central limit theorems, modules of continuity and diophantine type of irrational rotations, Journal d’Analyse Mathématique61 (1993), 47–59. · Zbl 0790.60027
[25] [La] J. Lamperti,Probability, Benjamin, New York, 1966.
[26] [Le] V. P. Leonov,On the dispersion of time means of a stationary stochastic process, Teoriya Veroyatnostei i Primeneniya6 (1961), 93–101 (in Russian; English translation in Theory of Probability and its Applications6 (1961), 87–93).
[27] [Li] M. Lin,On the uniform ergodic theorem, Proceedings of the American Mathematical Society43 (1974), 337–340. · Zbl 0252.47004
[28] [LiSi] M. Lin and R. Sine,Ergodic theory and the functional equation (I)x=y, Journal of Operator Theory10 (1983), 153–166. · Zbl 0553.47006
[29] [N] J. Neveu,Mathematical Foundations of the Calculus of Probability, Holden Day, San Francisco, 1965. · Zbl 0137.11301
[30] [P] K. Petersen,Ergodic Theory, Cambridge University Press, Cambridge, 1983.
[31] [RSz-N] F. Riesz and B. Sz-Nagy,Leçons d’analyse fonctionnelle, third edition, Akadémiai Kiadó, Budapest, 1955.
[32] [St] E. M. Stein,On the maximal ergodic theorem, Proceedings of the National Academy of Sciences of the United States of America47 (1961), 1894–1897. · Zbl 0182.47102
[33] [Y] K. Yosida,Functional Analysis, third edition, Springer, Berlin, 1971. · Zbl 0217.16001
[34] [Z] A. Zygmund,Trigonometric Series I–II, corrected second edition, Cambridge University Press, Cambridge, 1968.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.