PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
1995 | 114 | 2 | 127-145
Tytuł artykułu

Averages of unitary representations and weak mixing of random walks

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let S be a locally compact (σ-compact) group or semigroup, and let T(t) be a continuous representation of S by contractions in a Banach space X. For a regular probability μ on S, we study the convergence of the powers of the μ-average Ux = ʃ T(t)xdμ(t). Our main results for random walks on a group G are: (i) The following are equivalent for an adapted regular probability on G: μ is strictly aperiodic; $U^n$ converges weakly for every continuous unitary representation of G; U is weakly mixing for any ergodic group action in a probability space. (ii) If μ is ergodic on G metrizable, and $U^n$ converges strongly for every unitary representation, then the random walk is weakly mixing: $n^{-1} ∑_{k=1}^n |⟨μ^{k}*f,g⟩| → 0$ for $g ∈ L_∞(G)$ and $f ∈ L_{1}(G)$ with ʃ fdλ = 0. (iii) Let G be metrizable, and assume that it is nilpotent, or that it has equivalent left and right uniform structures. Then μ is ergodic and strictly aperiodic if and only if the random walk is weakly mixing. (iv) Weak mixing is characterized by the asymptotic behaviour of $μ^n$ on $UCB_{l}(G)$
Słowa kluczowe
Twórcy
autor
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel
  • Institut für Mathematische Stochastik, Lotzestrasse 13, Göttingen, Germany.
Bibliografia
  • [AaLWe] J. Aaronson, M. Lin and B. Weiss, Mixing properties of Markov operators and ergodic transformations, and ergodicity of cartesian products, Israel J. Math. 33 (1979), 198-224.
  • [A] R. Azencott, Espaces de Poisson des groupes localement compacts, Lecture Notes in Math. 148, Springer, 1970.
  • [D] Y. Derriennic, Lois "Zéros ou deux" pour les processus de Markov. Applications aux marches aléatoires, Ann. Inst. H. Poincaré B 12 (1976), 111-129.
  • [DGu] Y. Derriennic et Y. Guivarc'h, Théorème de renouvellement pour les groupes non-moyennables, C. R. Acad. Sci. Paris Sér. A 277 (1973), 613-615.
  • [DL_1] Y. Derriennic et M. Lin, Sur la tribu asymptotique de marches aléatoires sur les groupes, Séminaire de Probabilités, Rennes, 1983.
  • [DL_2] Y. Derriennic et M. Lin, Convergence of iterates of averages of certain operator representations and of convolution powers, J. Funct. Anal. 85 (1989), 86-102.
  • [DL_3] Y. Derriennic et M. Lin, On pointwise convergence in random walks, in: Almost Everywhere Convergence, Academic Press, 1989, 189-193.
  • [Di] J. Dixmier, Les C*-algèbres et leurs représentations, $2^e$ éd., Gauthier-Villars, Paris, 1969; English transl.: C*-algebras, North-Holland, 1977.
  • [Do] L. Dor, On sequences spanning a complex $l_1$ space, Proc. Amer. Math. Soc. 47 (1975), 515-516.
  • [F] S. R. Foguel, Ergodic Theory of Markov Processes, van Nostrand, 1969.
  • [G] S. Glasner, On Choquet-Deny measures, Ann. Inst. H. Poincaré B 12 (1976), 1-10.
  • [HRo] E. Hewitt and K. Ross, Abstract Harmonic Analysis, Vol. I, 2nd ed., Springer, Berlin, 1979.
  • [HoM] K. Hofmann and A. Mukherjea, Concentration functions and a class of non-compact groups, Math. Ann. 256 (1981), 535-548.
  • [JL_1] L. Jones and M. Lin, Ergodic theorems of weak mixing type, Proc. Amer. Math. Soc. 57 (1976), 50-52.
  • [JL_2] L. Jones and M. Lin, Unimodular eigenvalues and weak mixing, J. Funct. Anal. 35 (1980), 42-48.
  • [JRT] R. Jones, J. Rosenblatt and A. Tempelman, Ergodic theorems for convolutions of a measure on a group, Illinois J. Math. 38 (1994), 521-553.
  • [KaV] V. A. Kaimanovich and A. M. Vershik, Random walks on discrete groups: boundary and entropy, Ann. Probab. 11 (1983), 457-490.
  • [KeMa] J. Kerstan und K. Matthes, Gleichverteilungseigenschaften von Faltungpotenzen auf lokalkompakten Abelschen Gruppen, Wiss. Z. Univ. Jena 14 (1965), 457-462.
  • [K] U. Krengel, Ergodic Theorems, de Gruyter, Berlin, 1985.
  • [L] M. Lin, Ergodic properties of an operator obtained from a continuous representation, Ann. Inst. H. Poincaré B 13 (1977), 321-331.
  • [LW_1] M. Lin and R. Wittmann, Ergodic sequences of averages of group representations, Ergodic Theory Dynamical Systems 14 (1994), 181-196.
  • [LW_2] M. Lin and R. Wittmann, Convergence of representation averages and of convolution powers, Israel J. Math. 88 (1994), 125-157.
  • [Ly] Yu. Lyubich, Introduction to the Theory of Banach Representations of Groups, Birkhäuser, Basel, 1988.
  • [P] J. P. Pier, Amenable Locally Compact Groups, Wiley, New York, 1984.
  • [R] J. Rosenblatt, Ergodic and mixing random walks on locally compact groups, Math. Ann. 257 (1981), 31-42.
  • [Ros] H. P. Rosenthal, A characterization of Banach spaces containing $l_1$, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411-2413.
  • [S] A. J. Stam, On shifting iterated convolutions, Compositio Math. 17 (1966), 268-280.
  • [T] A. Tempelman, Ergodic Theorems for Group Actions, Kluwer, Dordrecht, 1992.
  • [Wi] G. Willis, The structure of totally disconnected locally compact groups, Math. Ann. 300 (1994), 341-363.
  • [Z] R. Zimmer, Ergodic Theory and Semi-simple Groups, Birkhäuser, Basel, 1984.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv114i2p127bwm
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.