Large deviation properties of weakly interacting processes via weak convergence methods

Annals of Probability

Volumn 40, Issue 1, 2012, Pages 74-102

  Budhiraja, Amarjit ^a   Dupuis, Paul ^b   Fischer, Markus ^b  

^a University of North Carolina at Chapel Hill   (United States)

^b BROWN UNIVERSITY   (United States)

Author keywords

Delay; Interacting random processes; Large deviations; Martingale problem; McKean Vlasov equation; Optimal stochastic control; Stochastic differential equation; Weak convergence

Indexed keywords

EID: 84865069712     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/10-AOP616     Document Type: Article

References (34)

1. Arous, G.B. and Guionnet, A. (1995). Large deviations for Langevin spin glass dynamics. Probab. Theory Related Fields 102 455-509. MR1346262
2. Ben Arous, G. and Brunaud, M. (1990). Méthode de Laplace: Étude variationnelle des fluctuations de diffusions de type "champ moyen." Stochastics Stochastics Rep. 31 79-144. MR1080535
3. Ben Arous, G. and Zeitouni, O. (1999). Increasing propagation of chaos for mean field models. Ann. Inst. H. Poincaré Probab. Statist. 35 85-102. MR1669916
4. Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York. MR1700749
5. Boué, M. and Dupuis, P. (1998). A variational representation for certain functionals of Brownian motion. Ann. Probab. 26 1641-1659. MR1675051
6. Budhiraja, A. and Dupuis, P. (2000). A variational representation for positive functionals of infinite dimensional Brownian motion. Probab. Math. Statist. 20 39-61. MR1785237
7. Budhiraja, A., Dupuis, P. and Maroulas, V. (2008). Large deviations for infinite dimensional stochastic dynamical systems. Ann. Probab. 36 1390-1420. MR2435853
8. Budhiraja, A., Dupuis, P. and Maroulas, V. (2011). Variational representations for continuous time processes. Ann. Inst. Henri Poincaré Probab. Stat. 47 725-747. MR2841073
9. Dawson, D.A. and Del Moral, P. (2005). Large deviations for interacting processes in the strong topology. In Statistical Modeling and Analysis for Complex Data Problems. GERAD 25th Anniv. Ser. 1 179-208. Springer, New York. MR2189537
10. Dawson, D.A. and Gärtner, J. (1987). Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics 20 247-308. MR0885876
11. Dawson, D.A. and Gärtner, J. (1994). Multilevel large deviations and interacting diffusions. Probab. Theory Related Fields 98 423-487. MR1271106
12. Del Moral, P. and Guionnet, A. (1998). Large deviations for interacting particle systems: Applications to non-linear filtering. Stochastic Process. Appl. 78 69-95. MR1653296
13. Djehiche, B. and Schied, A. (1998). Large deviations for hierarchical systems of interacting jump processes. J. Theoret. Probab. 11 1-24. MR1607396
14. Duan, J. and Millet, A. (2009). Large deviations for the Boussinesq equations under random influences. Stochastic Process. Appl. 119 2052-2081. MR2519356
15. Dudley, R. M. (2002). Real Analysis and Probability. Cambridge Studies in Advanced Mathematics 74. Cambridge Univ. Press, Cambridge. MR1932358
16. Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. Wiley, New York. MR1431744
17. Feng, J. and Kurtz, T. G. (2006). Large Deviations for Stochastic Processes. Mathematical Surveys and Monographs 131. Amer. Math. Soc., Providence, RI. MR2260560
18. Fontbona, J. (2004). Uniqueness for a weak nonlinear evolution equation and large deviations for diffusing particles with electrostatic repulsion. Stochastic Process. Appl. 112 119-144. MR2062570
19. Funaki, T. (1984). A certain class of diffusion processes associated with nonlinear parabolic equations. Z. Wahrsch. Verw. Gebiete 67 331-348. MR0762085
20. Gärtner, J. (1988). On the McKean-Vlasov limit for interacting diffusions. Math. Nachr. 137 197-248. MR0968996
21. Herrmann, S., Imkeller, P. and Peithmann, D. (2008). Large deviations and a Kramers' type law for self-stabilizing diffusions. Ann. Appl. Probab. 18 1379-1423. MR2434175
22. Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York. MR1121940
23. Kushner, H. J. (1990). Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems. Systems & Control: Foundations & Applications 3. Birkhäuser, Boston, MA. MR1102242
24. Léonard, C. (1995). Large deviations for long range interacting particle systems with jumps. Ann. Inst. H. Poincaré Probab. Statist. 31 289-323. MR1324810
25. Léonard, C. (1995). On large deviations for particle systems associated with spatially homogeneous Boltzmann type equations. Probab. Theory Related Fields 101 1-44. MR1314173
26. Méléard, S. (1996). Asymptotic behaviour of some interacting particle systems; McKeanVlasov and Boltzmann models. In Probabilistic Models for Nonlinear Partial Differential Equations (Montecatini Terme, 1995). Lecture Notes in Math. 1627 42-95. Springer, Berlin. MR1431299
27. Oelschläger, K. (1984). A martingale approach to the law of large numbers for weakly interacting stochastic processes. Ann. Probab. 12 458-479. MR0735849
28. Rachev, S. T. (1991). Probability Metrics and the Stability of Stochastic Models. Wiley, Chichester. MR1105086
29. Ren, J. and Zhang, X. (2005). Schilder theorem for the Brownian motion on the diffeomorphism group of the circle. J. Funct. Anal. 224 107-133. MR2139106
30. Röckner, M., Zhang, T. and Zhang, X. (2010). Large deviations for stochastic tamed 3D Navier-Stokes equations. Appl. Math. Optim. 61 267-285. MR2585144
31. Sritharan, S.S. and Sundar, P. (2006). Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise. Stochastic Process. Appl. 116 1636-1659. MR2269220
32. Stroock, D.W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 233. Springer, Berlin. MR0532498
33. Tanaka, H. (1984). Limit theorems for certain diffusion processes with interaction. In Stochastic Analysis (Katata/Kyoto, 1982). North-Holland Mathematical Library 32 469-488. North-Holland, Amsterdam. MR0780770
34. Yong, J. and Zhou, X. Y. (1999). Stochastic Controls: Hamiltonian systems and HJB equations. Applications of Mathematics (New York) 43. Springer, New York. MR1696772