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Asymptotic results for non-linear processes of the McKean tagged-molecule type

Published online by Cambridge University Press:  14 July 2016

Gaston Giroux
Gaston Giroux*
Affiliation:
Université de Sherbrooke
*
∗Postal address: Département de mathématiques et d'informatique, Université de Sherbrooke, Sherbrooke, PQ, J1K 2R1, Canada.
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Abstract

McKean's tagged-molecule process is a non-linear homogeneous two-state Markov chain in continuous time, constructed with the aid of a binary branching process. For each of a large class of branching processes we construct a similar process. The construction is carefully done and the weak homogeneity is deduced. A simple probability argument permits us to show convergence to the equidistribution (½, ½) and to note that this limit is a strong equilibrium. A non-homogeneous Markov chain result is also used to establish the geometric rate of convergence. A proof of a Boltzmann H-theorem is also established.

Keywords

BOLTZMANN H-THEOREMGEOMETRIC RATEBRANCHING PROCESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 23 , Issue 1 , March 1986 , pp. 42 - 51
Copyright
Copyright © Applied Probability Trust 1986 

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Footnotes

Research supported in part by NSERC, Canada, under Grant No A-5365 and in part by a grant of Formation de Chercheurs et Actions Concertées, Gouvernement du Québec.

References

Athreya, K. B. and Ney, P. (1982) Branching Processes. Springer-Verlag, Berlin.Google Scholar
Blumenthal, R. M. and Getoor, R. K. (1968) Markov Processes and Potential Theory. Academic Press, New York.Google Scholar
Chung, K. L. (1981) Lectures from Markov Processes to Brownian Motion. Springer-Verlag, Berlin.Google Scholar
Donsker, M. D. and Varadhan, S. R. S. (1983) Asymptotic evaluation of certain Markov process expectations for large time IV. Comm. Pure Appl. Math. 36, 183–212.Google Scholar
Dynkin, E. B. (1965) Markov Processes I, II. Springer-Verlag, Berlin.Google Scholar
Ellis, R. S. (1984) Large deviations for a general class of random vectors. Ann. Prob. 12, 1–12.Google Scholar
Ernst, M. H. (1981) Nonlinear model-Boltzmann equations and exact solutions. Phys. Rep. 78, 1–171.CrossRefGoogle Scholar
Giroux, G. (1983) Extension du théorème H de Boltzmann. Ann. Sci. Math. Québec. 7, 29–37.Google Scholar
Iscoe, I., Ney, P. and Nummelin, E. (1984) Large deviations of uniformly recurrent Markov additive processes. Report, Department of Mathematics, University of Helsinki.Google Scholar
Kullback, S. (1959) Information Theory and Statistics. Wiley, New York.Google Scholar
Kullback, S. and Leibler, R. A. (1951) On information and sufficiency. Ann. Math. Statist. 22, 79–86.Google Scholar
Mckean, H. P. Jr (1966a) A class of Markov processes associated with nonlinear parabolic equations. Proc. Nat. Acad. Sci. USA 56, 1907–1911.Google Scholar
Mckean, H. P. Jr (1966b) Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas. Arch. Rat. Mech. Anal. 21, 343–367.Google Scholar
Mckean, H. P. Jr (1967) An exponential formula for solving Boltzmann's equation for a Maxwellian gas. J. Combinatorial Theory 2, 358–382.Google Scholar
Meyer, P. A. (1967) Processus de Markov. Lecture Notes in Mathematics 26, Springer-Verlag, Berlin.Google Scholar
Seneta, E. (1981) Non-Negative Matrices. Springer-Verlag, Berlin.Google Scholar
Spohn, H. (1980) Kinetic equations from Hamiltonian dynamics: Markovian limits. Rev. Mod. Phys. 52, 569–615.CrossRefGoogle Scholar
Sznitman, A. S. (1982) Equations de types Boltzmann, spatialement homogènes. C.R. Acad. Sci. Paris 295, (I), 363–366.Google Scholar
Sznitman, A. S. (1984) Equations de types Boltzmann, spatialement homogènes. Z. Wahrscheinlichkeitsth. 66, 559–592.Google Scholar
Tanaka, H. (1978) Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrscheinlichkeitsth. 46, 67–105.Google Scholar
Wild, E. (1951) On Boltzmann's equation in the kinetic theory of gases. Proc. Camb. Phil. Soc. 47, 602–609.Google Scholar