doi: 10.3934/dcdsb.2021137
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Homogenization for stochastic reaction-diffusion equations with singular perturbation term

1. 

Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

2. 

School of Mathematics, Southeast University, Nanjing 211189, China

* Corresponding author: gaohj@hotmail.com

Received  August 2020 Early access May 2021

Fund Project: The second author is supported by NSFC Grant No. 11531006 and Science Climbing Program of Southeast University

The main purpose of this paper is to study the homogenization problem of stochastic reaction-diffusion equations with singular perturbation term. The difficulty in studying such problems is how to get the uniform estimates of the equations under the influence of the singularity term. Firstly, we use the properties of the elliptic equation corresponding to the generator to eliminate the influence of singular terms and obtain the uniform estimates of the slow equation and thus, get the tightness. Finally, we prove that under appropriate assumptions, the slow equation converges to a homogenization equation in law.

Citation: Yangyang Shi, Hongjun Gao. Homogenization for stochastic reaction-diffusion equations with singular perturbation term. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021137
References:
[1]

A. AnderssonM. HefterA. Jentzen and R. Kurniawan, Regularity properties for solutions of infinite dimensional Kolmogorov equations in Hilbert spaces, Potential Anal., 50 (2019), 347-379.  doi: 10.1007/s11118-018-9685-7.  Google Scholar

[2]

D. BlömkerM. Hairer and G. A. Pavliotis, Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities, Nonlinearity, 20 (2007), 1721-1744.  doi: 10.1088/0951-7715/20/7/009.  Google Scholar

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F. CampilloM. Kleptsyna and A. Piatnitski, Homogenization of random parabolic operator with large potential, Stochastic Process. Appl., 93 (2001), 57-85.  doi: 10.1016/S0304-4149(00)00095-8.  Google Scholar

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S. Cerrai and M. Freidlin, Averaging principle for a class of stochastic reaction-diffusion equations, Probab. Theory Related Fields, 144 (2009), 137-177.  doi: 10.1007/s00440-008-0144-z.  Google Scholar

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H. B. Fu and J. Q. Duan, An averaging principle for two-scale stochastic partial differential equations, Stoch. Dyn., 11 (2011), 353-367.  doi: 10.1142/S0219493711003346.  Google Scholar

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D. Gtarek and B. Goldys, On uniqueness in law of solutions to stochastic evolution equations in Hilbert spaces, Stochastic Anal. Appl., 12 (1994), 193-203.  doi: 10.1080/07362999408809346.  Google Scholar

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R. Z. Khasminskii, On the principle of averaging the Itô's stochastic differential equations, Kybernetika (Prague), 4 (1968), 260-279.   Google Scholar

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R. Z. Khasminskii, A limit theorem for solutions of differential equations with a random right hand part, Teor. Verojatnost. I Primenen, 11 (1966), 444-462.   Google Scholar

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T. G. Kurtz, A limit theorem for perturbed operator semigroups with applications to random evolutions, J. Functional Analysis, 12 (1973), 55-67.  doi: 10.1016/0022-1236(73)90089-X.  Google Scholar

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W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.  Google Scholar

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A. J. Majda, R. V. Abramov and M. J. Grote, Information Theory and Stochastics for Multiscale Nonlinear Systems, American Mathematical Society, Providence, RI, 2005. doi: 10.1090/crmm/025.  Google Scholar

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A. J. MajdaI. Timofeyev and E. V. Eijnden, A mathematical framework for stochastic climate models, Comm. Pure Appl. Math., 54 (2001), 891-974.  doi: 10.1002/cpa.1014.  Google Scholar

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E. Pardoux and A. L. Piatnitski, Homogenization of a nonlinear random parabolic partial differential equation, Stochastic Process. Appl., 104 (2003), 1-27.  doi: 10.1016/S0304-4149(02)00221-1.  Google Scholar

[28]

E. Pardoux and A. Y. Veretennikov, On the Poisson equation and diffusion approximation. I, Ann. Probab., 29 (2001), 1061-1085.  doi: 10.1214/aop/1015345596.  Google Scholar

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E. Pardoux and A. Y. Veretennikov, On Poisson equation and diffusion approximation. II, Ann. Probab., 31 (2003), 1166-1192.  doi: 10.1214/aop/1055425774.  Google Scholar

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[31]

G. A. Pavliotis and A. M. Stuart, Multiscale Methods. Averaging and Homogenization, Springer, New York, 2008.  Google Scholar

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S. Peszat and J. Zabczyk, Strong Feller property and irreducibility for diffusions on Hilbert spaces, Ann. Probab., 23 (1995), 157-172.   Google Scholar

[33]

K. Spiliopoulos, Fluctuation analysis and short time asymptotics for multiple scales diffusion processes, Stoch. Dyn., 14 (2014), 1350026 (22 pages). doi: 10.1142/S0219493713500263.  Google Scholar

[34]

W. Wang and A. J. Roberts, Average and deviation for slow-fast stochastic partial differential equations, J. Differential Equations, 253 (2012), 1265-1286.  doi: 10.1016/j.jde.2012.05.011.  Google Scholar

show all references

References:
[1]

A. AnderssonM. HefterA. Jentzen and R. Kurniawan, Regularity properties for solutions of infinite dimensional Kolmogorov equations in Hilbert spaces, Potential Anal., 50 (2019), 347-379.  doi: 10.1007/s11118-018-9685-7.  Google Scholar

[2]

D. BlömkerM. Hairer and G. A. Pavliotis, Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities, Nonlinearity, 20 (2007), 1721-1744.  doi: 10.1088/0951-7715/20/7/009.  Google Scholar

[3]

F. CampilloM. Kleptsyna and A. Piatnitski, Homogenization of random parabolic operator with large potential, Stochastic Process. Appl., 93 (2001), 57-85.  doi: 10.1016/S0304-4149(00)00095-8.  Google Scholar

[4]

S. Cerrai, Second Order PDE's in Finite and Infinite Dimension. A Probabilistic Approach, Springer-Verlag, Berlin, 2001. doi: 10.1007/b80743.  Google Scholar

[5]

S. Cerrai, A Khasminskii type averaging principle for stochastic reaction-diffusion equations, Ann. Appl. Probab., 19 (2009), 899-948.  doi: 10.1214/08-AAP560.  Google Scholar

[6]

S. Cerrai and M. Freidlin, Averaging principle for a class of stochastic reaction-diffusion equations, Probab. Theory Related Fields, 144 (2009), 137-177.  doi: 10.1007/s00440-008-0144-z.  Google Scholar

[7]

S. Cerrai and A. Lunardi, Averaging principle for nonautonomous slow-fast systems of stochastic reaction-diffusion equations: The almost periodic case, SIAM J. Math. Anal., 49 (2017), 2843-2884.  doi: 10.1137/16M1063307.  Google Scholar

[8]

A. ChauvièreL. Preziosi and C. Verdier, Cell Mechanics: From single scale-based models to multiscale modeling, Journal of Biological Dynamics, 8 (2014), 74-78.   Google Scholar

[9] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513.  Google Scholar
[10]

J. Q. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier, Amsterdam, 2014.  Google Scholar

[11]

S. N. Ethier and T. G. Kurtz, Markov Processes. Characterization and Convergence, John Wiley & Sons, Inc., New York, 1986. doi: 10.1002/9780470316658.  Google Scholar

[12]

H. B. Fu and J. Q. Duan, An averaging principle for two-scale stochastic partial differential equations, Stoch. Dyn., 11 (2011), 353-367.  doi: 10.1142/S0219493711003346.  Google Scholar

[13]

S. Gailus and K. Spiliopoulos, Discrete-time statistical inference for multiscale diffusions, Multiscale Model. Simul., 16 (2018), 1824-1858.  doi: 10.1137/17M1147408.  Google Scholar

[14]

G. A. Gottwald, D. T. Crommelin and C. L. E. Franzke, Stochastic climate theory, Nonlinear and Stochastic Climate Dynamics, 209-240, Cambridge Univ. Press, Cambridge, 2017.  Google Scholar

[15]

D. Gtarek and B. Goldys, On uniqueness in law of solutions to stochastic evolution equations in Hilbert spaces, Stochastic Anal. Appl., 12 (1994), 193-203.  doi: 10.1080/07362999408809346.  Google Scholar

[16]

W. Janke, Rugged Free Energy Landscapes: Common Computational Approaches to Spin Glasses, Structural Glasses and Biological Macromolecules, Springer-Verlag Berlin Heidelberg, 2008. Google Scholar

[17]

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer Berlin Heidelberg, 1994. doi: 10.1007/978-3-642-84659-5.  Google Scholar

[18]

R. Z. Khasminskii, On the principle of averaging the Itô's stochastic differential equations, Kybernetika (Prague), 4 (1968), 260-279.   Google Scholar

[19]

R. Z. Khasminskii, A limit theorem for solutions of differential equations with a random right hand part, Teor. Verojatnost. I Primenen, 11 (1966), 444-462.   Google Scholar

[20]

T. G. Kurtz, A limit theorem for perturbed operator semigroups with applications to random evolutions, J. Functional Analysis, 12 (1973), 55-67.  doi: 10.1016/0022-1236(73)90089-X.  Google Scholar

[21]

W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.  Google Scholar

[22]

A. J. Majda, R. V. Abramov and M. J. Grote, Information Theory and Stochastics for Multiscale Nonlinear Systems, American Mathematical Society, Providence, RI, 2005. doi: 10.1090/crmm/025.  Google Scholar

[23]

A. J. MajdaC. Franzke and B. Khouider, An applied mathematics perspective on stochastic modelling for climate, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366 (2008), 2429-2455.  doi: 10.1098/rsta.2008.0012.  Google Scholar

[24]

A. J. MajdaI. Timofeyev and E. V. Eijnden, A mathematical framework for stochastic climate models, Comm. Pure Appl. Math., 54 (2001), 891-974.  doi: 10.1002/cpa.1014.  Google Scholar

[25]

M. R. Morse and K. Spiliopoulos, Moderate deviations for systems of slow-fast diffusions, Asymptot. Anal., 105 (2017), 97-135.  doi: 10.3233/ASY-171434.  Google Scholar

[26]

E. Pardoux and R. Bouc, Asymptotic analysis of P.D.E.s with wide-band noise disturbances, and expansion of the moments, Stochastic Analysis and Applications, 2 (1984), 369-422.  doi: 10.1080/07362998408809044.  Google Scholar

[27]

E. Pardoux and A. L. Piatnitski, Homogenization of a nonlinear random parabolic partial differential equation, Stochastic Process. Appl., 104 (2003), 1-27.  doi: 10.1016/S0304-4149(02)00221-1.  Google Scholar

[28]

E. Pardoux and A. Y. Veretennikov, On the Poisson equation and diffusion approximation. I, Ann. Probab., 29 (2001), 1061-1085.  doi: 10.1214/aop/1015345596.  Google Scholar

[29]

E. Pardoux and A. Y. Veretennikov, On Poisson equation and diffusion approximation. II, Ann. Probab., 31 (2003), 1166-1192.  doi: 10.1214/aop/1055425774.  Google Scholar

[30]

E. Pardoux and A. Y. Veretennikov, On the Poisson equation and diffusion approximation. III, Ann. Probab., 33 (2005), 1111-1133.  doi: 10.1214/009117905000000062.  Google Scholar

[31]

G. A. Pavliotis and A. M. Stuart, Multiscale Methods. Averaging and Homogenization, Springer, New York, 2008.  Google Scholar

[32]

S. Peszat and J. Zabczyk, Strong Feller property and irreducibility for diffusions on Hilbert spaces, Ann. Probab., 23 (1995), 157-172.   Google Scholar

[33]

K. Spiliopoulos, Fluctuation analysis and short time asymptotics for multiple scales diffusion processes, Stoch. Dyn., 14 (2014), 1350026 (22 pages). doi: 10.1142/S0219493713500263.  Google Scholar

[34]

W. Wang and A. J. Roberts, Average and deviation for slow-fast stochastic partial differential equations, J. Differential Equations, 253 (2012), 1265-1286.  doi: 10.1016/j.jde.2012.05.011.  Google Scholar

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