Moderate deviation principles for weakly interacting particle systems. (English) Zbl 1369.60012
Summary: Moderate deviation principles for empirical measure processes associated with weakly interacting Markov processes are established. Two families of models are considered: the first corresponds to a system of interacting diffusions whereas the second describes a collection of pure jump Markov processes with a countable state space. For both cases the moderate deviation principle is formulated in terms of a large deviation principle (LDP), with an appropriate speed function, for suitably centered and normalized empirical measure processes. For the first family of models the LDP is established in the path space of an appropriate Schwartz distribution space whereas for the second family the LDP is proved in the space of \(l_2\) (the Hilbert space of square summable sequences)-valued paths. Proofs rely on certain variational representations for exponential functionals of Brownian motions and Poisson random measures.
MSC:
| 60F10 | Large deviations |
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 60J75 | Jump processes (MSC2010) |
| 60J60 | Diffusion processes |
| 60G57 | Random measures |
Keywords:
moderate deviations; large deviations; weakly interacting particle systems; jump-diffusions; nonlinear Markov processes; Laplace principle; variational representations; mean field asymptotics; Schwartz distributions; Poisson random measuresReferences:
| [1] | Baladron, J., Fasoli, D., Faugeras, O., Touboul, J.: Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and Fitzhugh-Nagumo neurons. J. Math. Neurosci. 2(1), 10 (2012) · Zbl 1322.60205 · doi:10.1186/2190-8567-2-10 |
| [2] | Borkar, V.S., Sundaresan, R.: Asymptotics of the invariant measure in mean field models with jumps. Stoch. Syst. 2(2), 322-380 (2012) · Zbl 1296.60258 · doi:10.1214/12-SSY064 |
| [3] | Borovkov, A.A., Mogul’skii, A.A.: Probabilities of large deviations in topological spaces. II. Sib. Math. J. 21(5), 653-664 (1980) · Zbl 0458.60019 · doi:10.1007/BF00973879 |
| [4] | Boué, M., Dupuis, P.: A variational representation for certain functionals of Brownian motion. Ann. Probab. 26(4), 1641-1659 (1998) · Zbl 0936.60059 · doi:10.1214/aop/1022855876 |
| [5] | Braun, W., Hepp, K.: The Vlasov dynamics and its fluctuations in the 1/n limit of interacting classical particles. Commun. Math. Phys. 56(2), 101-113 (1977) · Zbl 1155.81383 · doi:10.1007/BF01611497 |
| [6] | Budhiraja, A., Dupuis, P.: A variational representation for positive functionals of infinite dimensional Brownian motion. Probab. Math. Stat. 20(1), 39-61 (2000) · Zbl 0994.60028 |
| [7] | Budhiraja, A., Dupuis, P., Fischer, M.: Large deviation properties of weakly interacting processes via weak convergence methods. Ann. Probab. 40(1), 74-102 (2012) · Zbl 1242.60026 · doi:10.1214/10-AOP616 |
| [8] | Budhiraja, A., Dupuis, P., Fischer, M., Ramanan, K.: Limits of relative entropies associated with weakly interacting particle systems. Electron. J. Probab. 20(80), 1-22 (2015) · Zbl 1321.60202 |
| [9] | Budhiraja, A., Dupuis, P., Ganguly, A.: Moderate deviation principles for stochastic differential equations with jumps. Ann. Probab. (2015) (to appear) · Zbl 1346.60026 |
| [10] | Budhiraja, A., Dupuis, P., Maroulas, V.: Large deviations for infinite dimensional stochastic dynamical systems. Ann. Probab., 1390-1420 (2008) · Zbl 1155.60024 |
| [11] | Budhiraja, A., Dupuis, P., Maroulas, V.: Variational representations for continuous time processes. Ann. de l’Institut Henri Poincaré(B), Probab. et Stat. 47(3), 725-747 (2011) · Zbl 1231.60018 · doi:10.1214/10-AIHP382 |
| [12] | Budhiraja, A., Kira, E., Saha, S.: Central limit results for jump-diffusions with mean field interaction and a common factor. Math (2014). arXiv:1405.7682 · Zbl 1373.60051 |
| [13] | Budhiraja, A., Wu, R.: Some fluctuation results for weakly interacting multi-type particle systems. Stoch. Process. Appl. 126(8), 2253-2296 (2016) · Zbl 1338.60228 · doi:10.1016/j.spa.2016.01.010 |
| [14] | Chong, C., Klüppelberg, C.: Partial mean field limits in heterogeneous networks (2015). arXiv:1507.01905 (preprint) · Zbl 1427.60197 |
| [15] | Collet, F.: Macroscopic limit of a bipartite Curie-Weiss model: a dynamical approach. J. Stat. Phys. 157(6), 1301-1319 (2014) · Zbl 1310.82020 · doi:10.1007/s10955-014-1105-9 |
| [16] | Dawson, D.A.: Critical dynamics and fluctuations for a mean-field model of cooperative behavior. J. Stat. Phys. 31(1), 29-85 (1983) · doi:10.1007/BF01010922 |
| [17] | Dawson, D.A., Gärtner, J.: Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics 20(4), 247-308 (1987) · Zbl 0613.60021 · doi:10.1080/17442508708833446 |
| [18] | Del Moral, P., Hu, S., Wu, L.: Moderate deviations for interacting processes. Stat. Sinica 25(3), 921-951 (2015) · Zbl 1415.60032 |
| [19] | Dupuis, P., Ellis, R.S.: A weak convergence approach to the theory of large deviations, vol. 902. Wiley, New York (2011) · Zbl 0904.60001 |
| [20] | Dupuis, P., Ramanan, K., Wu, W.: Sample path large deviation principle for mean field weakly interacting jump processes (2012) (preprint) · Zbl 0996.60071 |
| [21] | Gel’fand, I.M., Vilenkin, N.Y.: Generalized functions, vol. 4. New York (1964) · Zbl 0136.11201 |
| [22] | Graham, C., Méléard, S.: Stochastic particle approximations for generalized Boltzmann models and convergence estimates. Ann. Probab. 25(1), 115-132 (1997) · Zbl 0873.60076 · doi:10.1214/aop/1024404281 |
| [23] | Hitsuda, M., Mitoma, I.: Tightness problem and stochastic evolution equation arising from fluctuation phenomena for interacting diffusions. J. Multivar. Anal. 19(2), 311-328 (1986) · Zbl 0604.60059 · doi:10.1016/0047-259X(86)90035-7 |
| [24] | Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Elsevier, London (1981) · Zbl 0495.60005 |
| [25] | Kallianpur, G., Xiong, J.: Stochastic differential equations in infinite dimensional spaces. Lect. Notes Monogr. Ser. 26, iii-342 (1995) · Zbl 0859.60050 |
| [26] | Kolokoltsov, V.N.: Nonlinear Markov Processes and Kinetic Equations, vol. 182. Cambridge University Press, Cambridge (2010) · Zbl 1222.60003 |
| [27] | Kurtz, T.G., Xiong, J.: Particle representations for a class of nonlinear SPDEs. Stoch. Process. Appl. 83(1), 103-126 (1999) · Zbl 0996.60071 · doi:10.1016/S0304-4149(99)00024-1 |
| [28] | Kurtz, T.G., Xiong, J.: A stochastic evolution equation arising from the fluctuations of a class of interacting particle systems. Commun. Math. Sci. 2(3), 325-358 (2004) · Zbl 1087.60050 · doi:10.4310/CMS.2004.v2.n3.a1 |
| [29] | Leonard, C.: Large deviations for long range interacting particle systems with jumps. Ann. de l’IHP Prob. et Stat. 31, 289-323 (1995) · Zbl 0839.60032 |
| [30] | McKean, H.P.: A class of Markov processes associated with nonlinear parabolic equations. Proc. Natl. Acad. Sci. USA 56(6), 1907-1911 (1966) · Zbl 0149.13501 · doi:10.1073/pnas.56.6.1907 |
| [31] | McKean, H.P.: Propagation of chaos for a class of non-linear parabolic equations. In: Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), pp. 41-57. Air Force Office Sci. Res., Arlington (1967) · Zbl 0936.60059 |
| [32] | Méléard, S.: Convergence of the fluctuations for interacting diffusions with jumps associated with Boltzmann equations. Stochastics 63(3-4), 195-225 (1998) · Zbl 0905.60073 |
| [33] | Mitoma, I.: Tightness of probabilities on C([0, 1]; \[{\cal S}^{\prime }S\]′) and D([0, 1]; \[{\cal S}^{\prime }S\]′). Ann. Probab. 11(4), 989-999 (1983) · Zbl 0527.60004 · doi:10.1214/aop/1176993447 |
| [34] | Oelschlager, K.: A martingale approach to the law of large numbers for weakly interacting stochastic processes. Ann. Probab. 12(2), 458-479 (1984) · Zbl 0544.60097 · doi:10.1214/aop/1176993301 |
| [35] | Shiga, T., Tanaka, H.: Central limit theorem for a system of Markovian particles with mean field interactions. Probab. Theory Relat. Fields 69(3), 439-459 (1985) · Zbl 0607.60095 |
| [36] | Spiliopoulos, K., Sowers, R.B.: Default clustering in large pools: large deviations. SIAM J. Financial Math. 6(1), 86-116 (2015) · Zbl 1397.60063 · doi:10.1137/130944060 |
| [37] | Sznitman, A.-S.: Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated. J. Funct. Anal. 56(3), 311-336 (1984) · Zbl 0547.60080 · doi:10.1016/0022-1236(84)90080-6 |
| [38] | Sznitman, A.S.: Topics in propagation of chaos. In: Ecole d’Eté de Probabilités de Saint-Flour XIX—1989, pp. 165-251. Springer, New York (1991) · Zbl 1338.60228 |
| [39] | Tanaka, H.: Limit theorems for certain diffusion processes with interaction. North Holl. Math. Libr. 32, 469-488 (1984) · Zbl 0552.60051 · doi:10.1016/S0924-6509(08)70405-7 |
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