×

Large deviations for interacting multiscale particle systems. (English) Zbl 1508.60036

The goal of this article is to obtain the large deviations principle (LDP) for interacting particle systems of diffusion type in multiscale environments.
The authors study the combined effect of weak mean field interactions in a fast oscillating multiscale environment from the point of view of large deviations for the empirical measure of the particles. They use methods from weak convergence and stochastic control [P. Dupuis and R. S. Ellis, A weak convergence approach to the theory of large deviations. Chichester: John Wiley & Sons (1997; Zbl 0904.60001)] and some methods for mean field stochastic control problems [K. Jańczak, Probab. Math. Stat. 28, No. 1, 41–47 (2008; Zbl 1147.60036)].
Their main results are given in Theorem 3.7, that gives the large deviations principle of the empirical distribution of the particles in the combined limit \(N \to \infty\) and \(\varepsilon \downarrow 0 \).
This result seems to extend some important results given in [D. A. Dawson and J. Gärtner, Stochastics 20, 247–308 (1987; Zbl 0613.60021); Dupuis and Ellis, loc. cit.; A. Budhiraja et al., Ann. Probab. 40, No. 1, 74–102 (2012; Zbl 1242.60026)] or [M. Röckner et al., Ann. Inst. Henri Poincaré, Probab. Stat. 57, No. 1, 547–576 (2021; Zbl 1491.60088)].

MSC:

60F10 Large deviations
60F05 Central limit and other weak theorems
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Almada, S. A.; Spiliopoulos, K., Scaling limits and exit law for multiscale diffusions, J. Asymptot. Anal., 87, 65-90 (2014) · Zbl 1370.37102
[2] Ambrosio, L.; Gigli, N.; Savaré, G., Gradient Flows: In Metric Spaces and in the Space of Probability Measures (2005), Springer: Springer NY · Zbl 1090.35002
[3] Ansari, A., Mean first passage time solution of the Smoluchowski equation: Application of relaxation dynamics in myoglobin, J. Chem. Phys., 112, 5, 2516-2522 (2000)
[4] Baldi, P., Large deviations for diffusions processes with homogenization and applications, Ann. Probab., 19, 2, 509-524 (1991) · Zbl 0735.60026
[5] Barbu, V.; Röckner, M., From nonlinear Fokker-Planck equations to solutions of distribution dependent SDE, Ann. Probab., 48, 4, 1902-1920 (2020) · Zbl 1469.60216
[6] Barré, J.; Bernardin, C.; Chétrite, R.; Chopra, Y.; Mariani, M., Gamma convergence approach for the large deviations of the density in systems of interacting diffusion processes (2019)
[7] Benedetto, D.; Caglioti, E.; Carrillo, J. A.; Pulvirenti, M., A non-Maxwellian steady distribution for one-dimensional granular media, J. Stat. Phys., 91, 979-990 (1998) · Zbl 0921.60057
[8] Bensoussan, A.; Lions, J. L.; Papanicolau, G., Asymptotic Analysis for Periodic Structures (1978), North Holland: North Holland Amsterdam · Zbl 0404.35001
[9] Bezemek, Z.; Spiliopoulos, K., Rate of homogenization for fully-coupled McKean-Vlasov SDEs (2022)
[10] Billingsley, P., Convergence of Probability Measures (1999), Wiley: Wiley NY · Zbl 0172.21201
[11] Binney, J.; Tremaine, S., Galactic Dynamics (2008), Princeton University Press: Princeton University Press Princeton · Zbl 1136.85001
[12] Borkar, V.; Gaitsgory, V., Averaging of singularly perturbed controlled stochastic differential equations, Appl. Math. Optim., 56, 2, 169-209 (2007) · Zbl 1139.93022
[13] Brunick, G.; Shreve, S., Mimicking an Itô process by a solution of a stochastic differential equation, Ann. Appl. Probab., 23, 4, 1584-1628 (2013) · Zbl 1284.60109
[14] Bryngelson, J. D.; Onuchic, J. N.; Socci, N. D.; Wolynes, P. G., Funnels, pathways and the energy landscape of protein folding: A synthesis, Proteins, 21, 3, 167-195 (1995)
[15] Budhirja, A.; Conroy, M., Asymptotic behavior of stochastic currents under large deviation scaling with mean field interaction and vanishing noise (2021)
[16] Budhirja, A.; Conroy, M., Empirical measure and small noise asymptotics under large deviation scaling for interacting diffusions, J. Theoret. Probab., 35, 295-349 (2022) · Zbl 1484.60028
[17] Budhirja, A.; Dupuis, P., A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist., 20 (2001)
[18] Budhirja, A.; Dupuis, P.; Fischer, M., Large devation properties of weakly interacting particles via weak convergence methods, T.A. Prob., 40, 74-100 (2012) · Zbl 1242.60026
[19] Carmona, R.; Delarue, F., Probabilistic Theory of Mean Field Games with Applications I (2018), Springer: Springer NY · Zbl 1422.91014
[20] Carmona, R.; Delarue, F.; Lacker, D., Mean field games with common noise, Ann. Probab., 44, 6, 3740-3803 (2016) · Zbl 1422.91083
[21] Claisse, J.; Talay, D.; Tan, X., A pseudo-Markov property for controlled diffusion processes, SIAM J. Control Optim., 54, 2, 1017-1029 (2016) · Zbl 1341.60097
[22] Dawson, D. A., Critical dynamics and fluctuations for a mean-field model of cooperative behavior, J. Stat. Phys., 31, 29-85 (1983)
[23] 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
[24] Dawson, D. A.; Gärtner, J., Large deviations, free energy functional and quasi-potential for a mean field model of interacting diffusions, Mem. Amer. Math. Soc., 78, 398 (1989) · Zbl 0677.60106
[25] Del Moral, P.; Niclas, A., A Taylor expansion of the square root matrix function, J. Math. Anal. Appl., 465, 1, 259-266 (2018) · Zbl 1401.15009
[26] Delarue, F.; Lacker, D.; Ramanan, K., From the master equation to mean field game limit theory: Large deviations and concentration of measure, Ann. Probab., 48, 1, 211-263 (2020) · Zbl 1445.60025
[27] Delgadino, M. G.; Gvalani, R. S.; Pavliotis, G. A., On the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitions, Arch. Ration. Mech. Anal., 241, 91-148 (2021) · Zbl 07364831
[28] Djete, F. M.; Possamaï, D.; Tan, X., Mckean-vlasov optimal control: Limit theory and equivalence between different formulations (2020)
[29] Djete, F. M.; Possamaï, D.; Tan, X., Mckean-Vlasov optimal control: The dynamic programming principle, Ann. Probab., 50, 2, 791-833 (2022) · Zbl 1491.49018
[30] Dudley, I. M., Real Analysis and Probability (2010), Cambridge University Press: Cambridge University Press Cambridge
[31] Dupuis, P.; Ellis, R. S., A Weak Convergence Approach to the Theory of Large Deviations (1997), Wiley: Wiley NY · Zbl 0904.60001
[32] Dupuis, P.; Spiliopoulos, K., Large deviations for multiscale diffusion via weak convergence methods, Stochastic Process. Appl., 122, 4, 1947-1987 (2012) · Zbl 1247.60034
[33] Dupuis, P.; Spiliopoulos, K.; Zhou, X., Escaping from an attractor: Importance sampling and rest points I, Ann. Appl. Probab., 25, 5, 2909-2958 (2015) · Zbl 1334.65007
[34] Ethier, S.; Kurtz, T., Markov Processes: Characterization and Convergence (1986), Wiley: Wiley NY · Zbl 0592.60049
[35] Feng, J.; Forde, M.; Fouque, J. P., Short-maturity asymptotics for a fast mean-reverting heston stochastic volatility model, SIAM J. Financial Math., 1, 1, 126-141 (2010) · Zbl 1203.91321
[36] Feng, J.; Fouque, J. P.; Kumar, R., Small-time asymptotics for fast mean-reverting stochastic volatility models, Ann. Appl. Probab., 22, 4, 1541-1575 (2012) · Zbl 1266.60049
[37] Feng, J.; Kurtz, G., Large Deviations for Stochastic Processes (2006), American Mathematical Society: American Mathematical Society Providence · Zbl 1113.60002
[38] Fischer, M., On the form of the large deviation rate function for the empirical measures of weakly interacting systems, Bernoulli, 20, 4, 1765-1801 (2014) · Zbl 1327.60070
[39] Fischer, M., On the connection between symmetric N-player games and mean field games, Ann. Appl. Probab., 27, 2, 757-810 (2017) · Zbl 1375.91009
[40] Fouque, J. P.; Papanicolaou, G.; Sircar, K. R., Derivatives in Financial Markets with Stochastic Volatility (2000), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0954.91025
[41] Freidlin, M.; Sowers, R., A comparison of homogenization and large deviations, with applications to wavefront propagation, Stoch. Process their Appl., 82, 1, 23-52 (1999) · Zbl 0996.60035
[42] Freidlin, M. I.; Wentzell, A. D., Random Perturbations of Dynamical Systems (2012), Springer: Springer Heidelberg · Zbl 1267.60004
[43] Gaitsgor, V.; Nguyen, M. T., Multiscale singularly perturbed control systems: Limit occupational measures sets and averaging, SIAM J. Control Optim., 41, 3, 954-974 (2002) · Zbl 1027.34071
[44] Ganguly, A.; Sundar, P., Inhomogeneous functionals and approximations of invariant distributions of ergodic diffusions: Central limit theorem and moderate deviation asymptotics, Stochastic Process. Appl., 133, 74-110 (2021) · Zbl 1469.60075
[45] Garnier, J.; Papanicolaou, G.; Yang, T. W., Large deviations for a mean field model of systemic risk, SIAM J. Financ. Math., 4, 1, 151-184 (2013) · Zbl 1283.60044
[46] Garnier, J.; Papanicolaou, G.; Yang, T. W., Consensus convergence with stochastic effects, Vietnam J. Math., 45, 1-2, 51-75 (2017) · Zbl 1372.92118
[47] Gyöngy, I., Mimicking the one-dimensional marginal distributions of processes having an Itô differential, Probab. Theory Related Fields, 71, 4, 501-516 (1986) · Zbl 0579.60043
[48] Hong, W.; Li, S.; Liu, W.; Sun, X., Central limit type theorem and large deviations for multi-scale McKean-Vlasov SDEs (2021)
[49] Hyeon, C.; Thirumalai, D., Can energy landscapes roughness of proteins and RNA be measured by using mechanical unfolding experiments?, Proc. Natl. Acad. Sci., 100, 18, 10249-10253 (2003)
[50] Isaacson, S. A.; Ma, J.; Spiliopoulos, K., Mean field limits of particle-based stochastic reaction-diffusion models, SIAM J. Math. Anal., 54, 1, 453-511 (2022) · Zbl 1482.35115
[51] Karatzas, I.; Shreve, S., Brownian Motion and Stochastic Calculus (1998), Springer: Springer NY
[52] Kolokolnikov, T.; Bertozzi, A.; Fetecau, R.; Lewis, M., Emergent behaviour in multi-particle systems with non-local interactions, Physica D, 260, 1-4 (2013)
[53] Kushner, H. J., Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems (1990), Birkhäuser: Birkhäuser Boston-Basel-Berlin · Zbl 0931.93003
[54] Lacker, D., Mean field games via controlled martingale problems: Existence of Markovian equilibria, Stochastic Process. Appl., 125, 7, 2856-2894 (2015) · Zbl 1346.60083
[55] Lacker, D., Limit theory for controlled McKean-Vlasov dynamics, SIAM J. Control Optim., 55, 3, 1641-1672 (2017) · Zbl 1362.93167
[56] Lipster, R., Large deviations for two scaled diffusions, Probab. Theory Related Fields, 106, 1, 71-104 (1996) · Zbl 0855.60030
[57] Lućon, E.; Stannat, W., Transition from Gaussian to non-Gaussian fluctuations for mean-field diffusions in spatial interaction, Ann. Probab., 26, 6, 3840-3909 (2016) · Zbl 1358.60104
[58] Majda, A. J.; Franzke, C.; Khouider, B., An applied mathematics perspective on stochastic modelling for climate, Phil. Trans. R. Soc. A, 336, 1875, 2429-2455 (2008) · Zbl 1153.86315
[59] Morse, M. R.; Spiliopoulos, K., Moderate deviations principle for systems of slow-fast diffusions, Asymptot. Anal., 105, 3-4, 97-135 (2017) · Zbl 1390.60111
[60] Motsch, S.; Tadmor, E., Heterophilious dynamics enhances consensus, SIAM Rev., 56, 4, 577-621 (2014) · Zbl 1310.92064
[61] Collective dynamics from bacteria to crowds: An excursion through modeling, analysis and simulation, (Muntean, A., CISM International Centre for Mechanical Sciences. Courses and Lectures, vol. 533 (2014), Springer: Springer Vienna)
[62] Øksendal, B., Stochastic Differential Equations (2003), Springer: Springer NY · Zbl 1025.60026
[63] Pardoux, E.; Veretennikov, A. Y., On Poisson equation and diffusion approximation I, Ann. Probab., 29, 3, 1061-1085 (2001) · Zbl 1029.60053
[64] Pardoux, E.; Veretennikov, A. Y., On Poisson equation and diffusion approximation II, Ann. Probab., 31, 3, 1166-1192 (2003) · Zbl 1054.60064
[65] Pavliotis, G.; Stuart, G. A., Multiscale Methods (2008), Springer: Springer NY · Zbl 1160.35006
[66] Rachev, S. T., Probability Metrics and the Stability of Stochastic Models (1991), Wiley: Wiley NY · Zbl 0744.60004
[67] Röckner, M.; Sun, X.; Xie, Y., Strong convergence order for slow-fast McKean-Vlasov stochastic differential equations, Ann. Inst. H. Poincaré Probab. Statist., 57, 1, 547-576 (2021) · Zbl 1491.60088
[68] Röckner, M.; Xie, L., Diffusion approximation for fully coupled stochastic differential equations, Ann. Probab., 49, 3, 101-122 (2021)
[69] Spiliopoulos, K., Large deviations and importance sampling for systems of slow-fast motion, Appl. Math. Optim., 67, 123-161 (2013) · Zbl 1259.93136
[70] Spiliopoulos, K., Fluctuation analysis and short time asymptotics for multiple scales diffusion processes, Stoch. Dyn., 14, 3, Article 1350026 pp. (2014) · Zbl 1291.60050
[71] Spiliopoulos, K., Quenched large deviations for multiscale diffusion processes in random environments, Electron. J. Probab., 20, 15, 1-29 (2015) · Zbl 1320.60081
[72] Yamada, T.; Watanabe, S., On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ., 11, 1, 155-167 (1971) · Zbl 0236.60037
[73] Yu Veretennikov, A., On large deviations for SDEs with small diffusion and averaging, Stochastic Process. Appl., 89, 1, 69-79 (2000) · Zbl 1045.60065
[74] Yu Veretennikov, A., On large deviations in the averaging principle for SDEs with a “full dependence”, correction (2005), Initial article in Annals of Probability, Vol. 27, No. 1, (1999) 284-296 · Zbl 0939.60012
[75] Zwanzig, R., Diffusion in a rough potential, Proc. Natl. Acad. Sci., 85, 2029-2030 (1988)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.