Large Deviation Principle for Multi-Scale Stochastic Systems with Monotone Coefficients

Abstract

The Freidlin–Wentzell’s large deviation principle is established for a class of multi-scale stochastic models involving slow–fast components, where the slow component has general monotone drift and the fast component has dissipative drift driven by multiplicative noise. Our result is applicable to various slow–fast stochastic dynamical systems such as stochastic porous media equations, stochastic p-Laplace equations and stochastic reaction–diffusion equations. The weak convergence method and the technique of time discretization are used to establish the Laplace principle (equivalently, large deviation principle) under the multi-scale framework.

Keywords

SPDE / Multi-scale system / Large deviation principle / Weak convergence method / 60H15 / 60F10

References

[1] Ansari A. Mean first passage time solution of the Smoluchowski equation: application of relaxation dynamics in myoglobin. J. Chem. Phys.. 2000, 112: 2516-2522.

[2] Barbu, V., Da Prato, G., Röckner, M.: Stochastic Porous Media Equations. Lecture Notes in Math, vol. 2163. Springer, New York (2016)

[3] Bertram R, Rubin JE. Multi-timescale systems and fast-slow analysis. Math. Biosci.. 2017, 287: 105-121.

[4] Bessaih H, Millet A. Large deviation principle and inviscid shell models. Electron. J. Probab.. 2009, 14: 2551-2579.

[5] Bryc W. Pinsky M. Large deviations by the asymptotic value method. Diffusion Processes and Related Problems in Analysis. 1990, Boston, Birkhäuser447472

[6] Brzeźniak Z, Goldys B, Jegaraj T. Large deviations and transitions between equilibria for stochastic Landau–Lifshitz–Gilbert equation. Arch. Ration. Mech. Anal.. 2017, 226: 497-558.

[7] Budhiraja A, Dupuis P. A variational representation for positive functionals of infinite dimensional Brownian motion. Probab. Math. Stat.. 2000, 20: 39-61

[8] Budhiraja A, Dupuis P, Maroulas V. Large deviations for infinite dimensional stochastic dynamical systems. Ann. Probab.. 2008, 36: 1390-1420.

[9] Chen Y, Gao H. Well-posedness and large deviations for a class of SPDEs with Lévy noise. J. Differ. Equ.. 2017, 263: 5216-5252.

[10] Chueshov I, Millet A. Stochastic 2D hydrodynamical type systems: well posedness and large deviations. Appl. Math. Optim.. 2010, 61: 379-420.

[11] Dareiotis K, Gess B, Tsatsoulis P. Ergodicity for stochastic porous media equations with multiplicative noise. SIAM J. Math. Anal.. 2020, 52: 4524-4564.

[12] Dembo A, Zeitouni O. Large Deviations Techniques and Applications. 2000, New York, Springer

[13] Dupuis P, Ellis R. A Weak Convergence Approach to the Theory of Large Deviations. 1997, New York, Wiley.

[14] E, W., Engquist, B.: Multiscale modeling and computations. Not. Am. Math. Soc. 50, 1062–1070 (2003)

[15] E, W., Liu, D., Vanden-Eijnden, E.: Analysis of multiscale methods for stochastic differential equations. Commun. Pure Appl. Math. 58, 1544–1585 (2005)

[16] Freidlin MI. Random perturbations of reaction–diffusion equations: the quasideterministic approximation. Trans. Am. Math. Soc.. 1988, 305: 665-697

[17] Freidlin MI, Wentzell AD. Random Perturbations of Dynamical Systems, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 260. 1984, New York, Springer

[18] Gess B. Optimal regularity for the porous medium equation. J. Eur. Math. Soc.. 2021, 23: 425-465.

[19] Harvey E, Kirk V, Wechselberger M, Sneyd J. Multiple timescales, mixed mode oscillations and canards in models of intracellular calcium dynamics. J. Nonlinear Sci.. 2011, 21: 639-683.

[20] Hong, W., Li, M., Li, S., Liu, W.: Large deviations and averaging for stochastic tamed 3D Navier–Stokes equations with fast oscillations. Appl. Math. Optim. 86, 15 (2022)

[21] Hong W, Li S, Liu W. Freidlin–Wentzell type large deviation principle for multi-scale locally monotone SPDEs. SIAM J. Math. Anal.. 2021, 53(6): 6517-6561.

[22] Hong W, Li S, Liu W. Strong convergence rates in averaging principle for slow-fast McKean–Vlasov SPDEs. J. Differ. Equ.. 2022, 316: 94-135.

[23] Hong, W., Li, S., Liu, W., Sun, X.: Central limit type theorem and large deviations for multi-scale McKean–Vlasov SDEs. Probab. Theory Relat. Fields (in press)

[24] Kiefer Y. Averaging and Climate Models. In Stochastic Climate Models. 2000, Boston, Birkhäuser

[25] Liptser R. Large deviations for two scaled diffusions. Probab. Theory. Relat. Fields. 1996, 106: 71-104.

[26] Liu W. Harnack inequality and applications for stochastic evolution equations with monotone drifts. J. Evol. Equ.. 2009, 9: 747-770.

[27] Liu W. Large deviations for stochastic evolution equations with small multiplicative noise. Appl. Math. Optim.. 2010, 61: 27-56.

[28] Liu W, Röckner M. SPDE in Hilbert space with locally monotone coefficients. J. Funct. Anal.. 2010, 259: 2902-2922.

[29] Liu W, Röckner M. Local and global well-posedness of SPDE with generalized coercivity conditions. J. Differ. Equ.. 2013, 254: 725-755.

[30] Liu W, Röckner M. Stochastic Partial Differential Equations: An Introduction. 2015, Berlin, Springer.

[31] Liu, W., Röckner, M., Sun, X., Xie, Y.: Strong averaging principle for slow-fast stochastic partial differential equations with locally monotone coefficients. Appl. Math. Optim. 87, 39 (2023)

[32] Manna U, Sritharan SS, Sundar P. Large deviations for the stochastic shell model of turbulence. NoDEA Nonlinear Differ. Equ. Appl.. 2009, 16: 493-521.

[33] Matoussi A, Sabbagh W, Zhang T. Large deviation principles of obstacle problems for quasilinear stochastic PDEs. Appl. Math. Optim.. 2021, 83: 849-879.

[34] Pavliotis GA, Stuart AM. Multiscale Methods: Averaging and Homogenization, Texts in Applied Mathematics 53. 2008, New York, Springer

[35] Puhalskii AA. On large deviations of coupled diffusions with time scale separation. Ann. Probab.. 2016, 44: 3111-3186.

[36] Pukhalskii AA. On the theory of large deviations. Theory Probab. Appl.. 1993, 38: 490-497.

[37] Ren J, Zhang X. Freidlin-Wentzell’s large deviations for stochastic evolution equations. J. Funct. Anal.. 2008, 254: 3148-3172.

[38] Sritharan SS, Sundar P. Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise. Stochastic Process. Appl.. 2006, 116: 1636-1659.

[39] Sun, X., Wang, R., Xu, L., Yang, X.: Large deviation for two-time-scale stochastic Burgers equation, Stoch. Dyn. 21(5), Paper No. 2150023, 37 pp (2021)

[40] Varadhan SRS. Asymptotic probabilities and differential equations. Commun. Pure Appl. Math.. 1966, 19: 261-286.

[41] Varadhan SRS. Large deviations and Applications, CBMS-NSF Series in Applied Mathematics. 1984, Philadelphia, SIAM. 46

[42] Veretennikov AY. On large deviations for SDEs with small diffusion and averaging. Stochastic Process. Appl.. 2000, 89: 69-79.

[43] Wang W, Roberts AJ, Duan J. Large deviations and approximations for slow-fast stochastic reaction-diffusion equations. J. Differ. Equ.. 2012, 253: 3501-3522.

[44] Wu F, Tian T, Rawlings JB, Yin G. Approximate method for stochastic chemical kinetics with two-time scales by chemical Langevin equations. J. Chem. Phys.. 2016, 144. 174112

[45] Xiong J, Zhai J. Large deviations for locally monotone stochastic partial differential equations driven by Lévy noise. Bernoulli. 2018, 24: 2842-2874.

[46] Zhai J, Zhang T. Large deviations for stochastic models of two-dimensional second grade fluids. Appl. Math. Optim.. 2017, 75: 471-498.

Funding

National Natural Science Foundation of China(12171208, 11831014, 12090011)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature