Journal of Theoretical Probability

, Volume 11, Issue 1, pp 1–24 | Cite as

Large Deviations for Hierarchical Systems of Interacting Jump Processes

  • Boualem Djehiche
  • Alexander Schied
Article

Abstract

We investigate the large deviations principle from the McKean–Vlasov limit for a collection of jump processes obeying a two-level hierarchy interaction. A large deviation upper bound is derived and it is shown that the associated rate function admits a Lagrangian representation as well as a nonvariational one. Moreover, it is proved that the admissible paths for the weak solution of the McKean–Vlasov equation enjoy certain strong differentiability properties.

Two-level hierarchical interaction McKean–Vlasov limit large deviations Orlicz space measure-valued processes weak interaction epidemic process 

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REFERENCES

  1. 1.
    Billingsley, P. (1968). Convergence of Probability Measures, John Wiley and Sons, Inc., New York.Google Scholar
  2. 2.
    Dawson, D. A. (1984). Asymptotic analysis of multilevel stochastic systems. In Stochastic Differential Systems, Métivier, M. and Pardoux, E. (eds.), Lecture Notes in Control and Information Sciences 69, 76–89.Google Scholar
  3. 3.
    Dawson, D. A. (1985). Stochastic ensembles and hierarchies. In Stochastic Processes and their Applications, Itô, I. and Hida, T. (eds.), Lecture Notes in Math. 1203, 20–37.Google Scholar
  4. 4.
    Dawson, D. A. and Gärtner, J. (1994). Multilevel large deviations and interacting diffusions, Prob. Th. Rel. Fields 98, 432–487.Google Scholar
  5. 5.
    Dawson, D. A. and Gärtner, J. (1987). Large deviations from the McKean-Vlasov limit for weakly interacting diffusions, Stochastics 20, 247–308.Google Scholar
  6. 6.
    Dembo, A. and Zeitouni, O. (1993). Large Deviations Techniques, Jones and Bartlett Publishers, Boston, London.Google Scholar
  7. 7.
    Djehiche, B. and Kaj, I. (1995). The rate function for some measure-valued jump processes, Ann. Prob. 23, 1414–1438.Google Scholar
  8. 8.
    Ethier, S. and Kurtz, T. (1986). Markov Processes, Characterization and Convergence, John Wiley-Interscience, New York.Google Scholar
  9. 9.
    Hille, E. and Phillips, R. (1957). Functional Analysis and Semi-groups, American Mathematical Society, Providence.Google Scholar
  10. 10.
    Jakubowski, A. (1986). On the Skorohod topology, Ann. Inst. Henri Poincaré B22, 263–285.Google Scholar
  11. 11.
    Léonard, C. (1995). On the large deviations for particle systems associated with spatially homogeneous Boltzmann type equations, Prob. Th. Rel. Fields 101, 1–44.Google Scholar
  12. 12.
    Léonard, C. (1995). Large deviations for long range interacting particle systems with jumps, Ann. Inst. Henri Poincaré B31, 289–323.Google Scholar
  13. 13.
    Parthasarathy K. R. (1967). Probability Measures on Metric Spaces, Academic Press, New York.Google Scholar
  14. 14.
    Pukhalskii, A. (1994). The method of stochastic exponentials for large deviations, Stoch. Process. Applications 54, 45–70.Google Scholar
  15. 15.
    Rao, M. M. and Ren, Z. D. (1991). Theory of Orlicz Spaces, Marcel Dekker, Inc. New York.Google Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • Boualem Djehiche
    • 1
  • Alexander Schied
    • 2
  1. 1.Division of Mathematical StatisticsThe Royal Institute of TechnologyStockholmSweden
  2. 2.Institut für MathematikHumboldt-UniversitätBerlinGermany

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