Large Deviations of Fractional Stochastic FitzHugh–Nagumo Systems on

{\mathbb{R}}^n

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Abstract

This paper deals with the large deviation principle of the fractional stochastic FitzHugh–Nagumo systems on

{\mathbb{R}}^n

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {R}^n$$\end{document}

with superlinear drift. We first establish the well-posedness and uniform tail-estimates of solutions to the controlled system associated with the original stochastic equations. We then prove the strong convergence of solutions to the controlled system with respect to the weak topology of controls by the method of tail-ends estimates in order to overcome the non-compactness of Sobolev embeddings on unbounded domains. We finally show the large deviation principle of the FitzHugh–Nagumo system by the weak convergence method.

Keywords

Fractional stochastic FitzHugh–Nagumo system / Unbounded domain / Superlinear drift / Large deviation / Tail-estimates / Weak convergence method / 60F10 / 60H15 / 37L55 / 35R60

References

[1] Abe S, Thurner S. Anomalous diffusion in view of Einstein’s 1905 theory of Brownian motion. Physica A, 2005, 356: 403-407

[2] Adili A, Wang B. Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing. Discrete Contin. Dyn. Syst. B, 2013, 18(3): 643-666

[3] Alouini B. Finite dimensional global attractor for a fractional Schrödinger type equation with mixed anisotropic dispersion. J. Dyn. Diff. Equat., 2022, 34: 1237-1268

[4] Bao J, Yuan C. Large deviations for neutral functional SDEs with jumps. Stochastics, 2015, 87: 48-70

[5] 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

[6] Brzeźniak Z, Peng X, Zhai J. Well-posedness and large deviations for 2D stochastic Navier-Stokes equations with jumps. J. Eur. Math. Soc., 2023, 25: 3093-3176

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

[8] Budhiraja A, Dupuis P, Fischer M. Large deviation properties of weakly interacting processes via weak convergence methods. Ann. Probab., 2012, 40(1): 74-102

[9] Budhiraja A, Dupuis P, Maroulas V. Variational representations for continuous time processes. Ann. Inst. Henri Poincaré Probab. Stat., 2011, 47(3): 725-747

[10] Cerrai S, Debussche A. Large deviations for the two-dimensional stochastic Navier-Stokes equation with vanishing noise correlation. Ann. Inst. Henri Poincaré Probab. Stat., 2019, 55: 211-236

[11] Cerrai S, Röckner M. Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term. Ann. Probab., 2004, 32: 1100-1139

[12] Chen CN, Hu X. Stability analysis for standing pulse solutions to FitzHugh-Nagumo equations. Calc. Var. Partial. Differ. Equ., 2014, 49: 827-845

[13] Chen Z, Sun X, Wang B. Invariant measures and large deviation principles for stochastic Schrödinger delay lattice systems. Proc. Roy. Soc. Edinb. A, 2025, 155(5): 1813-1854

[14] Chen Z, Wang B. Invariant measures of fractional stochastic delay reaction-diffusion equations on unbounded domains. Nonlinearity, 2021, 34: 3969-4016

[15] Chen Z, Wang B. Limit measures and ergodicity of fractional stochastic reaction-diffusion equations on unbounded domains. Stoch. Dyn., 2022, 22(2): 2140012

[16] Chow PL. Large deviation problem for some parabolic Itô equations. Commun. Pure Appl. Math., 1992, 45: 97-120

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

[18] Cordoni F, Persio LD. Optimal control of the FitzHugh-Nagumo stochastic model with nonlinear diffusion. Appl. Math. Optim., 2021, 84(3): 2947-2968

[19] Dembo A, Zeitouni A. Large Deviations Techniques and Applications, 1998, Berlin. Springer-Verlag

[20] Dong Z, Zhai J, Zhang R. Large deviation principles for 3D stochastic primitive equations. J. Differential Equations, 2017, 263(5): 3110-3146

[21] Duan J, Millet A. Large deviations for the Boussinesq equations under random influences. Stoch. Process. Appl., 2009, 119: 2052-2081

[22] Dupuis P, Ellis RS. A Weak Convergence Approach to the Theory of Large Deviations, 1997, New York. Wiley-Interscience

[23] Eichinger K, Gnann MV, Kuehn C. Multiscale analysis for traveling-pulse solutions to the stochastic FitzHugh-Nagumo equations. Ann. Appl. Probab., 2022, 32(5): 3229-3282

[24] Fang S, Zhang T. A study of a class of stochastic differential equations with non-Lipschitzian coefficients. Probab. Theory Relat. Fields, 2005, 132: 356-390

[25] FitzHugh R. Impulses and physiological states in theoretical models of nerve membrane. Biophys. J., 1961, 1: 445-466

[26] Freidlin MI, Wentzell AD. Random Perturbations of Dynamical Systems, 2012, New York. Springer-Verlag

[27] Gal C, Warma M. Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions. Discrete Contin. Dyn. Syst., 2016, 36(3): 1279-1319

[28] Gu A, Li D, Wang B, Yang H. Regularity of random attractors for fractional stochastic reaction-diffusion equations on Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{R} ^n$$\end{document}. J. Differential Equations, 2018, 264: 7094-7137

[29] Gu A, Wang B. Random attractors of Fitzhugh-Nagumo systems driven by colored noise on unbounded domains. Stoch. Dyn., 2019, 19(5): 1950035

[30] Guan Q, Ma Z. Reflected symmetric α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document}-stable processes and regional fractional Laplacian. Probab. Theory Relat. Fields, 2006, 134: 649-694

[31] Hodgkin AL, Huxley AF. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol., 1952, 117: 500-544

[32] 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., 2022, 86(2): 15

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

[34] Jara M. Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps. Commun. Pure Appl. Math., 2009, 62: 198-214

[35] Jones CKRT. Stability of the travelling wave solution of the FitzHugh-Nagumo system. Trans. Amer. Math. Soc., 1984, 286: 431-469

[36] Kallianpur G, Xiong J. Large deviations for a class of stochastic partial differential equations. Ann. Probab., 1996, 24: 320-345

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

[38] Liu L, Caraballo T. Well-posedness and dynamics of a fractional stochastic integro-differential equation. Physica D, 2017, 355: 45-57

[39] Liu W, Song Y, Zhai J, Zhang T. Large and moderate deviation principles for McKean-Vlasov SDEs with jumps. Potential Anal., 2023, 59(3): 1141-1190

[40] Lu H, Bates P, Lu S, Zhang M. Dynamics of 3D fractional complex Ginzburg-Landau equation. J. Differential Equations, 2015, 259: 5276-5301

[41] Lv Y, Wang W. Limit dynamics for the stochastic FitzHugh-Nagumo system. Nonlinear Anal.: RWA, 2010, 11(4): 3091-3105

[42] Matoussi A, Sabbagh W, Zhang T. Large deviation principles of obstacle problems for quasilinear stochastic PDEs. Appl. Math. Optim., 2021, 83(2): 849-879

[43] Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE. 50, 2061–2070 (1962)

[44] Rauch J, Smoller J. Qualitative theory of the Fitzhugh-Nagumo equations. Adv. Math., 1978, 27: 12-44

[45] Röckner M, Schmuland B, Zhang X. Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions. Cond. Matt. Phys., 2008, 2: 247-259

[46] Röckner M, Zhang T, Zhang X. Large deviations for stochastic tamed 3D Navier-Stokes equations. Appl. Math. Optim., 2010, 61: 267-285

[47] Shi L, Wang R, Lu K, Wang B. Asymptotic behavior of stochastic FitzHugh-Nagumo systems on unbounded thin domains. J. Differential Equations, 2019, 267(7): 4373-4409

[48] Skrzypek L, You Y. Dynamics and synchronization of boundary coupled FitzHugh-Nagumo neural networks. Appl. Math. Comput., 2021, 388125545

[49] Sowers RB. Large deviations for a reaction-diffusion equation with non-Gaussian perturbations. Ann. Probab., 1992, 20: 504-537

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

[51] Sun X, Wang R, Xu L, Yang X. Large deviation for two-time-scale stochastic Burgers equation. Stoch. Dyn., 2021, 21(5): 2150023

[52] Tang BQ. Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains. Discrete Contin. Dyn. Syst., 2015, 35(1): 441-466

[53] Wang B. Pullback attractors for the non-autonomous FitzHugh-Nagumo system on unbounded domains. Nonlinear Anal.: TMA, 2009, 70(11): 3799-3815

[54] Wang B. Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations. Nonlinear Anal., 2017, 158: 60-82

[55] Wang B. Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise. J. Differential Equations, 2019, 268: 1-59

[56] Wang B. Large deviations of fractional stochastic equations with non-Lipschitz drift and multiplicative noise on unbounded domains. J. Differential Equations, 2023, 376: 1-38

[57] Wang R, Guo B, Wang B. Well-posedness and dynamics of fractional FitzHugh-Nagumo systems on RN\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{R} ^N$$\end{document} driven by nonlinear noise. Sci China Math, 2021, 64(11): 2395-2436

[58] Wu W, Zhai J. Large deviations for stochastic porous media equation on general measure spaces. J. Differential Equations, 2020, 269(11): 10002-10036

[59] Xu J, Miao Y, Liu J. Strong averaging principle for two-time-scale non-autonomous stochastic FitzHugh-Nagumo system with jumps. J. Math. Phys., 2016, 57(9): 092704

[60] Zhai J, Zhang T. Large deviations for stochastic models of two-dimensional second grade fluids. Appl. Math. Optim., 2017, 75(3): 471-498

Funding

National Natural Science Foundation of China(11971260, 12471167)

RIGHTS & PERMISSIONS

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