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Comparison of interacting diffusions and an application to their ergodic theory
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  • Published: December 1996

Comparison of interacting diffusions and an application to their ergodic theory

  • J. Theodore Cox1,
  • Klaus Fleischmann2 &
  • Andreas Greven3 

Probability Theory and Related Fields volume 105, pages 513–528 (1996)Cite this article

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  • 34 Citations

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Summary

A general comparison argument for expectations of certain multitime functionals of infinite systems of linearly interacting diffusions differing in the diffusion coefficient is derived. As an application we prove clustering occurs in the case when the symmetrized interaction kernel is recurrent, and the components take values in an interval bounded on one side. The technique also gives an alternative proof of clustering in the case of compact intervals.

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References

  • [BCG94] M. Bramson, J.T. Cox, A. Greven: Ergodicity of critical spatial branching processes in low dimensions. Ann. Probab., 21: 1946–1957 (1994)

    Google Scholar 

  • [BCGdH95] J. Baillon, P. Clement, A. Greven, F. den Hollander: On the attracting orbit of a nonlinear transformation arising from renormalization of hierarchically interacting diffusions: The noncompact case. Preprint, 1995.

  • [CG94] J.T. Cox, A. Greven: Ergodic theorems for infinite systems of locally interacting diffusions. Ann. Probab., 22(2): 833–853 (1994)

    Google Scholar 

  • [CGS95a] J.T. Cox, A. Greven, T. Shiga: Finite and infinite systems of interacting diffusions. Probab. Theory Relat. Fields 103, 165–197 (1995)

    Google Scholar 

  • [CGS95b] J.T. Cox, A. Greven, T. Shiga: Finite and infinite systems of interacting diffusions, part 2. Technical report, Syracuse Univ., 1994.

  • [Daw77] D.A. Dawson: The critical measure diffusion process. Z. Wahrsch. verw. Gebiete, 40: 125–145 (1977)

    Google Scholar 

  • [Deu94] J.-D. Deuschel: AlgebraicL 2 decay of attractive critical processes on the lattice. Ann. Probab., 22(1):264–283 (1994)

    Google Scholar 

  • [DFFP86] D.A. Dawson, K. Fleischmann, R.D. Foley, L.A. Peletier: A critical measure-valued branching process with infinite mean. Stoch. Analysis Appl., 4:117–129 (1986)

    Google Scholar 

  • [EK86] S.N. Ethier, T.G. Kurtz: Markov Processes: Characterization and Convergence. Wiley, New York, 1986.

    Google Scholar 

  • [FG94] K. Fleischmann, A. Greven: Diffusive clustering in an infinite system of hierarchically interacting diffusions. Probab. Theory Relat. Fields, 98: 517–566 (1994)

    Google Scholar 

  • [FG96] K. Fleischmann, A. Greven: Time-space analysis of the cluster-formation in interacting diffusions. Electronic J. Probab., 1: 1–46 (1996)

    Google Scholar 

  • [GS69] I.I. Gikhman, A.V. Skorohod: Introduction to the theory of random processes. W.B. Sauncers Co., Philadelphia, 1969.

    Google Scholar 

  • [Kle95] A. Klenke: Different clustering regimes in systems of hierarchically interacting diffusions. Ann. Probab. To appear

  • [Lig85] T.M. Liggett: Interacting Particle Systems. Springer-Verlag, New York, 1985.

    Google Scholar 

  • [LS81] T.M. Liggett, F. Spitzer: Ergodic theorems for coupled random walks and other systems with locally interacting components. Z. Wahrsch. verw. Gebiete, 56: 443–468 (1981)

    Google Scholar 

  • [RY91] D. Revuz, M. Yor: Continuous martingales and Brownian motion. Springer, Berlin, Heidelberg, New York, 1991.

    Google Scholar 

  • [Shi80] T. Shiga: An interacting system in population genetics. J. Mat. Kyoto Univ., 20: 213–242 (1980)

    Google Scholar 

  • [Shi92] T. Shiga: Ergodic theorems and exponential decay of sample paths for certain interacting diffusion systems. Osaka J. Math., 29:789–807 (1992)

    Google Scholar 

  • [SS80] T. Shiga, A. Shimizu: Infinite-dimensional stochastic differential equations and their applications. J. Mat. Kyoto Univ., 20: 395–416 (1980)

    Google Scholar 

  • [ZMAS88] Ya.B. Zeldovich, S.A. Molchanov, Ruzmaikin A.A, D.D. Sokoloff: Intermittency, diffusion and generation in a nonstationary random medium. Soviet Sci. Rev. Math. Phys., 7:1–110 (1988)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematics Department, Syracuse University, 13244, Syracuse, NY, USA

    J. Theodore Cox

  2. Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS), Mohrenstr. 39, D-10117, Berlin, Germany

    Klaus Fleischmann

  3. Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstr. 1 1/2, D-91054, Erlangen, Germany

    Andreas Greven

Authors
  1. J. Theodore Cox
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  2. Klaus Fleischmann
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  3. Andreas Greven
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Cox, J.T., Fleischmann, K. & Greven, A. Comparison of interacting diffusions and an application to their ergodic theory. Probab. Th. Rel. Fields 105, 513–528 (1996). https://doi.org/10.1007/BF01191911

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  • Received: 07 December 1994

  • Revised: 28 February 1996

  • Issue Date: December 1996

  • DOI: https://doi.org/10.1007/BF01191911

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Mathematics Subject Classification (1991)

  • 60K35
  • 60J60
  • 60J15
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