Probability Theory and Related Fields

, Volume 85, Issue 1, pp 13–26 | Cite as

Ergodicity of reversible reaction diffusion processes

  • Wan-Ding Ding
  • Richard Durrett
  • Thomas M. Liggett


Reaction-diffusion processes were introduced by Nicolis and Prigogine, and Haken. Existence theorems have been established for most models, but not much is known about ergodic properties. In this paper we study a class of models which have a reversible measure. We show that the stationary distribution is unique and is the limit starting from any initial distribution.


Stochastic Process Probability Theory Diffusion Process Stationary Distribution Mathematical Biology 
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  1. Boldrighini, C., DeMasi, A,, Pellegrinotti, A.: Non-equilibrium fluctuations in particle systems modelling diffusion-reaction systems (preprint 1989)Google Scholar
  2. Boldrighini, C., DeMasi, A., Pellegrinotti, A., Presutti, E.: Collective phenomena in interacting particle systems. Stoch. Proc. Appl.25, 137–152 (1987)CrossRefGoogle Scholar
  3. Chen, M.F., Infinite dimensional reaction diffusion processes. Acta Math. Sin. New Ser.1, 261–273 (1985)Google Scholar
  4. Chen, M.F.: Coupling for jump processes. Acta Math. Sin., New Ser.2, 123–126 (1986a)Google Scholar
  5. Chen, M.F.: Jump processes and particle systems. (In Chinese) Beijing Normal Univ. Press (1986b)Google Scholar
  6. Chen, M.F.: Existence theorems for interacting particle systems with non-compact state space. Sci. Sin., Ser. A,30, 148–156 (1987)Google Scholar
  7. Chen, M.F.: Stationary distributions for infinite particle systems with noncompact state space. Acta Math. Sci.9, 9–19 (1989)Google Scholar
  8. Dewel, G., Borckmans, P., Walgraef, D.: Nonequilibrium phase transitions and chemical instabilities. J. Stat. Phys.24, 119–137 (1981)Google Scholar
  9. Dewel, G., Walgraef, D., Borckmans, P.: Renormalization group approach to chemical instabilities. Z. Phys. B28, 235–237 (1977)Google Scholar
  10. Ding, W.D. and Zheng, X.G.: Ergodic theorems for linear growth processes with diffusion (preprint 1987)Google Scholar
  11. Feng, S. and Zheng, X.G.: Solutions of a class of nonlinear master equations. Carleton University Preprint no. 115 (1988)Google Scholar
  12. Feistel, R.: Nonlinear chemical reactions in diluted solutions In: Ebeling, W., Ulbricht, H. (eds.) Selforganization by nonlinear irreversible processes Proceedings, Mühlungsborn 1985. (Springer Ser. Synergetics, vol. 33) Berlin Heidelberg New York: Springer 1985Google Scholar
  13. Grassberger, P.: On phase transitions in Schlögl's second model. Z. Phys. B58, 229–244 (1982)Google Scholar
  14. Grassberger, P., Torre, A. de la: Reggeon field theory (Schlögl's first model) on a lattice: Monte Carlo calculations of critical behavior. Ann. Phys.122, 373–396 (1979)Google Scholar
  15. Haken, H.: Synergetics. Berlin Heidelberg New York: Springer 1977Google Scholar
  16. Hanuse, P.: Fluctuations in non-equilibrium phase transitions: critical behavior. In: Vidal, C., Pacault, A. (eds.) Nonlinear phenomena in chemical dynamics. Proceedings, Bordeaux 1981 (Springer Ser. Synergetics, vol. 12) Berlin Heidelberg New York: Springer 1981Google Scholar
  17. Hanuse, P., Blanché, A.: Simulation study of the critical behavior of a chemical model system. In: Garrido, L. (ed.) Systems far from equilibrium. Conference, Barcelona 1980. (Lect. Notes Phys., vol. 132, pp. 337–344) Berlin Heidelberg New York: Springer 1980Google Scholar
  18. Holley, R.: Free energy in a Markovian model of a lattice spin system. Commun. Math. Phys.23, 87–99 (1971)Google Scholar
  19. Holley, R.: An ergodic theorem for interacting systems with attractive interactions. Z. Wahrscheinlichkeitstheor. Verw. Geb.24, 325–334 (1972)Google Scholar
  20. Holley, R., Stroock, D.: A martingale approach to infinite systems of interacting particles. Ann. Probab.4, 195–228 (1976)Google Scholar
  21. Holley, R., Stroock, D.: In one and two dimensions every stationary measure for a stochastic Ising model is a Gibbs state. Commun. Math. Phys.55, 37–45 (1977)Google Scholar
  22. Janssen, H.K.: Stochastisches Reaktionsmodell für einen Nichtgleichgewichts-Phasenübergang. Z. Phys.270, 57–73 (1974)Google Scholar
  23. Janssen, H.K.: On the nonequilibrium phase transition in reaction diffusion systems with an absorbing stationary state. Z. Phys. B42, 151–154 (1981)Google Scholar
  24. Liggett, T.M.: An infinite particle systems with zero range interactions. Ann. Probab.1, 240–253 (1973)Google Scholar
  25. Liggett, T.M.: Interacting Particle Systems. Berlin Heidelberg New York: Springer 1985Google Scholar
  26. Liggett, T.M., Spitzer, F.: Ergodic theorems for coupled random walks and other systems with locally interacting components. Z. Wahrscheinlichkeitstheor. Verw. Geb.56, 443–468 (1981)Google Scholar
  27. Mountford, T.: The ergodicity of a class of reaction diffusion processes (preprint 1989)Google Scholar
  28. Neuhauser, C.: Untersuchung des Einflusses von Wanderung auf nichtlineare Dynamiken bein Ising-Modell und bein Schlöglmodell. Diplomarbeit thesis, Heidelberg (1988)Google Scholar
  29. Neuhauser, C.: An ergodic theorem for Schlögl models with small migration. Probab. Th. Rel. Fields85, 27–32 (1990)Google Scholar
  30. Nicolis, G., Priogogine, I.: Self-organization in nonequilibrium systems. New York: Wiley 1977Google Scholar
  31. Ohtsuki, T., Keyes, T.: Nonequilibrium critical phenomena in one component reaction diffusion systems. Phys. Rev. A35, 2697–2703 (1987)Google Scholar
  32. Schlögl, F.: Chemical reaction models and non-equilibrium phase transitions. Z. Phys.253, 147–161 (1972)Google Scholar
  33. Shiga, T.: Stepping stone models in population genetics and population dynamics. In: Albeverio, S., et al. (eds.) Stochastic processes in physics and engineering. pp. 345–355. Dordrecht: Reidel 1988Google Scholar
  34. Zheng, X.G., Ding, W.D.: Existence theorems for linear growth processes with diffusion. Acta Math. Sci.7, 25–42 (1987).Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Wan-Ding Ding
    • 1
  • Richard Durrett
    • 2
  • Thomas M. Liggett
    • 3
  1. 1.Department of MathematicsAnhui Normal UniversityWuhuPeople's Republic of China
  2. 2.Department of MathematicsCornell UniversityIthacaUSA
  3. 3.Department of MathematicsU.C.L.A.Los AngelesUSA

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