Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-27T23:58:30.942Z Has data issue: false hasContentIssue false
  • English
  • Français

Fluctuation theory for systems of signed and unsigned particles with interaction mechanisms based on intersection local times

Published online by Cambridge University Press:  01 July 2016

Robert J. Adler
Robert J. Adler*
Affiliation:
Technion—Israel Institute of Technology
*
∗Postal address: Faculty of Industrial Engineering and Management, Technion—Israel Institute of Technology, Technion City, Haifa 32000, Israel.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider two distinct models of particle systems. In the first we have an infinite collection of identical Markov processes starting at random throughout Euclidean space. In the second a random sign is associated with each process. An interaction mechanism is introduced in each case via intersection local times, and the fluctuation theory of the systems studied as the processes become dense in space. In the first case the fluctuation theory always turns out to be Gaussian, regardless of the order of the intersections taken to introduce the interaction mechanism. In the second case, an interaction mechanism based on kth order intersections leads to a fluctuation theory akin to a :φ k: Euclidean quantum field theory. We consider the consequences of these results and relate them to different models previously studied in the literature.

Keywords

MARKOV PROCESSESINTERACTING PARTICLESGAUSSIAN CHAOSEUCLIDEAN QUANTUM FIELD THEORIES
Type
Research Article
Information
Advances in Applied Probability , Volume 21 , Issue 2 , June 1989 , pp. 334 - 356
Copyright
Copyright © Applied Probability Trust 1989 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research supported in part by U.S. Air Force Office of Scientific Research, AFOSR 87–0298.

References

Adler, R. J. (1989) The net charge process for interacting, signed diffusions. Ann. Prob. 17.Google Scholar
Adler, R. J. and Epstein, R. (1987) Some central limit theorems for Markov paths and some properties of Gaussian random fields. Stoch. Proc. Appl. 24, 157–202.Google Scholar
Briemont, J., Kuroda, K. and Lebowitz, J. L. (1984) The structure of Gibbs states and phase coexistence for nonsymmetric continuum Widom Rowlinson models. Z. Wahrsheinlichkeitsth. 67, 121–138.Google Scholar
Dawson, D. (1983) Critical dynamics and fluctuations for a mean field model of cooperative behavior. J. Statist. Phys. 31, 29–85.CrossRefGoogle Scholar
Dynkin, E. B. (1981) Additive functionals of several time reverisble Markov processes. J. Funct. Anal. 42, 64–101.CrossRefGoogle Scholar
Dynkin, E. B. (1984) Gaussian and non-Gaussian random fields associated with Markov processes. J. Funct. Anal. 55, 344–376.CrossRefGoogle Scholar
Dynkin, E. B. and Mandelbaum, A. (1983) Symmetric statistics, Poisson point processes and multiple Wiener integrals. Ann. Statist. 11, 739–745.Google Scholar
Ellis, R. S. and Newman, C. M. (1978a) Fluctuations in Curie Weiss exemplis. In Proc. Internat. Conf. Mathematical Problems in Theoretical Physics. Lecture Notes in Physics 80, Springer-Verlag, Berlin.Google Scholar
Ellis, R. S. and Newman, C. M. (1978b) Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrsheinlichkeitsth. 44, 117–139.Google Scholar
Ellis, R. S. and Newman, C. M. (1978C) The statistics of Curie Weiss models. J. Statist. Phys. 19, 149–161.Google Scholar
Epstein, R. (1989) Some limit theorems for functionals of the Brownian sheet. Ann. Prob. 17.CrossRefGoogle Scholar
Glimm, J. and Jaffe, A. (1982) Quantum Physics: a Functional Integral Point of View. Springer-Verlag, Berlin.Google Scholar
Liggett, T. M. (1985) Interacting Particle Systems. Springer-Verlag, New York.Google Scholar
Major, P. (1981) Multiple Wiener–Ito Integrals. Lecture Notes in Mathematics 849, Springer-Verlag, Berlin.Google Scholar
Mckean, H. P. (1967) Propagation of chaos for a class of nonlinear parabolic equations. Lecture Series in Differential Equations 7, 41–57, Catholic Univ. of Amer. Washington, DC.Google Scholar
Nelson, E. (1966) A quartic interaction in two dimensions. In Mathematical Theory of Elementary Particles, ed. Goodman, R. and Segal, I.. MIT Press, Cambridge, Mass.Google Scholar
Rosen, J. (1983) A local time approach to the self intersections of Brownian paths in space. Commun. Math. Phys. 88, 327–338.Google Scholar
Shiga, T. and Tanaka, H. (1985) Central limit theorem for a system of Markovian particles with mean field interactions. Z. Wahrsheinlichkeitsth. 69, 439–459.Google Scholar
Simon, B. (1974) The ?(f2) Euclidean (Quantum) Field Theory . Princeton University Press, Princeton, New Jersey.Google Scholar
Snitzman, A. S. (1982) Equations de type boltzmann spatialement homogénes. C.R. Acad. Sci. Paris, Ser A 295, 363–366.Google Scholar
Snitzman, A. S. (1984) Non-linear reflecting diffusion process, and the propagation of chaos and fluctuations associated. J. Funct. Anal. 56, 311–336.Google Scholar
Snitzman, A. S. (1986) A propagation of chaos result for Burger's equation. Prob. Theory Rel. Fields 71, 581–613.Google Scholar
Symanzik, K. (1969) Euclidean quantum field theory. In Local Quantum Field Theory, ed. Jost, R.. Academic Press, New York.Google Scholar
Tanaka, H. (1982) Some probabilistic problems in the spatially homogeneous Boltzmann equation Proc. IFIP-ISI Conf. on Theory and Appl. of Random Fields, Bangladore.Google Scholar
Tanaka, H. (1984) Limit theorems for certain diffusion processes with interaction. Proc. Taniguchi International Symposium on Stochastic Analysis, ed. Ito, K.. Kinokuniya, Tokyo 469–488.Google Scholar
Tanaka, H. and Hitsuda, M. (1981) Central limit theorems for a simple diffusion model of interacting particles. Hiroshima Math. J. 11, 415–423.Google Scholar
Varadhan, S. (1969) Appendix to Euclidean quantum field theory, by K. Symanzik. In Local Quantum Field Theory, ed. Jost, R.. Academic Press, New York.Google Scholar
Wolpert, R. (1978a) Wiener path intersections and local time. J. Funct. Anal. 30, 329–340.Google Scholar
Wolpert, R. (1978b) Local time and a particle picture picture for Euclidean field theory. J. Funct. Anal. 30, 341–357.Google Scholar