Summary
Let (ξN) be a sequence of random variables with values in a topological space which satisfy the large deviation principle. For eachM and eachN, let ΞM, N denote the empirical measure associated withM independent copies of ξN. As a main result, we show that (ΞM, N) also satisfies the large deviation principle asM,N→∞. We derive several representations of the associated rate function. These results are then applied to empirical measure processes ΞM, N(t) =M −1 Σ Ni=1 δξ N i (t) 0≦t≦T, where (ξ N1 ,..., ξ N M (t)) is a system of weakly interacting diffusions with noise intensity 1/N. This is a continuation of our previous work on the McKean-Vlasov limit and related hierarchical models ([4], [5]).
References
Azencott, R.: Grandes deviations et applications. In: Ecole d'Eté de Probabilités de Saint-Flour VIII-1978. (Lect. Notes Math., Vol. 774, pp. 2–176) Berlin Heidelberg New York: Springer 1980
Billingsley, P.: Convergence of probability measures. New York: Wiley 1968
Crandall, M. G., Lions, P.-L.: Viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc.277, 1–42 (1983)
Dawson, D. A., Gärtner, J.: Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics20, 247–308 (1987)
Dawson, D. A., Gärtner, J.: Long time behaviour of interacting diffusions. In: Norris, J. R. (ed.) Stochastic calculus in application. (Proc. Cambridge Symp., 1987, pp. 29–54) Harlow Essex London New York: Longman 1988
Dawson, D. A., Gärtner, J.: Large deviations, free energy functional and quasi-potential for a mean field model of interacting diffusions. Mem. Am. Math. Soc.78, (1989)
Ekeland, I., Temam, R.: Convex analysis and variational problems. Amsterdam Oxford: North Holland 1976
Freidlin, M. I., Wentzell, A. D.: Random perturbations of dynamical systems. Berlin Heidelberg New York: Springer 1984
Gärtner, J.: On the McKean-Vlasov limit for interacting diffusions. Math. Nachr.137, 197–248 (1988)
Kellerer, H. G.: Duality theorems for marginal problems. Z. Wahrscheinlichkeitstheor. Verw. Geb.67, 399–432 (1984)
Lions, P.: Generalized solutions of Hamilton-Jacobi equations. Boston, London, Melbourne: Pitman 1982
Postnikov, M. M.: Introduction to Morse theory (in Russian) Moscow: Nauka 1971
Scheutzow, M.: Uniqueness and non-uniqueness of solutions of Vlasov-McKean equations. J. Aust. Math. Soc. Ser. A43, 246–256 (1987)
Schwartz, L.: Radon measures on arbitrary topological spaces and cylindrical measures. Oxford: Oxford University Press 1973
Stroock, D. W., Varadhan, S. R. S.: Multidimensional diffusion processes. Berlin Heidelberg New York: Springer 1979
Topsøe, F.: Topology and measure. (Lect. Notes Math., vol. 133) Berlin Heidelberg New York: Springer 1979
Varadhan, S. R. S.: Large deviations and applications. CBMS-NSF Regional Conference Series in Appl. Math., vol. 46 Philadelphia: SIAM 1984
Wentzell, A. D. Limit theorems on large deviations for Markov stochastic processes. Dordrecht: Kluwer Academic Publisher 1990
Wets, R. J. B.: Convergence of convex functions, variational inequalities and convex optimization problems. In: Cottle, R. W., Giannessi, F., Lions, J. L. (eds.) Variational inequalities and complementarity problems, pp. 375–403. Chichester: Wiley 1980
Author information
Authors and Affiliations
Additional information
Research partially supported by a Natural Science and Engineering Research Council of Canada operating grant
Rights and permissions
About this article
Cite this article
Dawson, D.A., Gärtner, J. Multilevel large deviations and interacting diffusions. Probab. Th. Rel. Fields 98, 423–487 (1994). https://doi.org/10.1007/BF01192835
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01192835
Mathematics Subject Classification
- 60F10
- 60k35
- 60J60