Summary
We consider a class of reaction-diffusion processes with state space NZd. The reaction part is described by a birth and death process where the rates are given by certain polynomials. The diffusion part is an irreducible symmetric random walk. We prove ergodicity in the case of a sufficiently small migration rate. For the proof we couple two processes and show that the density of the discrepancies goes to zero.
Article PDF
Similar content being viewed by others
References
Chen, M.F.: Infinite dimensional reaction diffusion processes. Acta Math. Sin., New Ser.1, 261–273 (1985)
Chen, M.F.: Existence theorems for interacting particle systems with non-compact state space. Sci. Sin. Ser., A.30, 148–156 (1987)
Ding, W.D., Durrett, R., and Liggett, T.: Ergodicity of reversible reaction diffusion processes. Probab. Th. Rel. Fields.85, 13–26 (1990)
Haken, H.: Synergetics. Berlin Heidelberg New York: Springer 1977
Janssen, H.K.: Stochastisches Reaktionsmodell für einen Nichtgleichgewichtsphasenübergang. Z. Phys.270, 67–73 (1974)
Liggett, T.: An infinite particle system with zero range interactions. Ann. Probab.1, 240–253 (1973)
Liggett, T.: Interacting particle systems. Berlin Heidelberg New York: Springer 1985
Nicolis, G., Prigogine, I.: Self-organization in nonequilibrium, systems. New York: Wiley 1977
Schlögl, F.: Chemical reaction models for phase transitions. Z. Phys.253, 147–161 (1972)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Neuhauser, C. An ergodic theorem for Schlögl models with small migration. Probab. Th. Rel. Fields 85, 27–32 (1990). https://doi.org/10.1007/BF01377625
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01377625