Ergodic theorems for coupled random walks and other systems with locally interacting components

  • Thomas M. Liggett
  • Frank Spitzer


In [5] the second author introduced a variety of new infinite systems with locally interacting components. On the basis of computations for the finite analogues of these systems, he made conjectures ragarding their limiting behavior as t→∞. This paper is devoted to the construction of these processes and to the proofs of these conjectures. We restrict ourselves primarily to spatially homogeneous situations; interesting problems remain unsolved in inhomogeneous cases. Two features distinguish these processes from most other infinite particle systems which have been studied. One is that the state spaces of these systems are noncompact; the other that even though the invariant measures are not generally of product form, one can nevertheless compute explicitly the first and second moments of the number of particles per site in equilibrium. The second moment computations are of inherent interest of course, and they play an important role in the proofs of the ergodic theorems as well.


State Space Stochastic Process Probability Theory Invariant Measure Product Form 
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  1. 1.
    Griffeath, D.: Annihilating and coalescing random walks on Z d. Z. Wahrscheinlichkeitstheorie verw. Gebiete 46, 55–65 (1978)Google Scholar
  2. 2.
    Holley, R., Liggett, T.M.: Generalized Potlatch and smoothing processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 55, 165–196 (1981)Google Scholar
  3. 3.
    Liggett, T.M.: A characterization of the invariant measures for an infinite particle system with interactions. Trans. Amer. Math. Soc. 179, 433–453 (1973)Google Scholar
  4. 4.
    Liggett, T.M.: An infinite particle system with zero range interactions. Ann. Probability 1, 240–253 (1973)Google Scholar
  5. 5.
    Spitzer, F.: Infinite systems with locally interacting components. [Ann. Probabilty; to appear in 1981]Google Scholar
  6. 6.
    Hsiao, C.T.: Stochastic processes with Gaussian interaction of components. To appear in Z. Wahrscheinlichkeitstheorie verw. Gebiete 1981]Google Scholar
  7. 7.
    Roussignol, M.: Un processus de saut sur IR a une infinité de particules. Ann. Inst. H. Poincaré, 16, 101–108 (1980Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • Thomas M. Liggett
    • 1
  • Frank Spitzer
    • 2
  1. 1.University of CaliforniaLos AngelesUSA
  2. 2.Dept. of MathematicsCornell UniversityIthacaUSA

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