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April  2022, 27(4): 2401-2426. doi: 10.3934/dcdsb.2021137

## 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 Published  April 2022 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 and Continuous Dynamical Systems - B, 2022, 27 (4) : 2401-2426. doi: 10.3934/dcdsb.2021137
##### References:
 [1] A. Andersson, M. Hefter, A. 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. [2] D. Blömker, M. 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. [3] F. Campillo, M. 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. [4] S. Cerrai, Second Order PDE's in Finite and Infinite Dimension. A Probabilistic Approach, Springer-Verlag, Berlin, 2001. doi: 10.1007/b80743. [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. [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. [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. [8] A. Chauvière, L. Preziosi and C. Verdier, Cell Mechanics: From single scale-based models to multiscale modeling, Journal of Biological Dynamics, 8 (2014), 74-78. [9] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513. [10] J. Q. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier, Amsterdam, 2014. [11] S. N. Ethier and T. G. Kurtz, Markov Processes. Characterization and Convergence, John Wiley & Sons, Inc., New York, 1986. doi: 10.1002/9780470316658. [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. [13] S. Gailus and K. Spiliopoulos, Discrete-time statistical inference for multiscale diffusions, Multiscale Model. Simul., 16 (2018), 1824-1858.  doi: 10.1137/17M1147408. [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. [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. [16] W. Janke, Rugged Free Energy Landscapes: Common Computational Approaches to Spin Glasses, Structural Glasses and Biological Macromolecules, Springer-Verlag Berlin Heidelberg, 2008. [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. [18] R. Z. Khasminskii, On the principle of averaging the Itô's stochastic differential equations, Kybernetika (Prague), 4 (1968), 260-279. [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. [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. [21] W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4. [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. [23] A. J. Majda, C. 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. [24] A. J. Majda, I. 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. [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. [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. [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. [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. [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. [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. [31] G. A. Pavliotis and A. M. Stuart, Multiscale Methods. Averaging and Homogenization, Springer, New York, 2008. [32] S. Peszat and J. Zabczyk, Strong Feller property and irreducibility for diffusions on Hilbert spaces, Ann. Probab., 23 (1995), 157-172. [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. [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.

show all references

##### References:
 [1] A. Andersson, M. Hefter, A. 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. [2] D. Blömker, M. 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. [3] F. Campillo, M. 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. [4] S. Cerrai, Second Order PDE's in Finite and Infinite Dimension. A Probabilistic Approach, Springer-Verlag, Berlin, 2001. doi: 10.1007/b80743. [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. [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. [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. [8] A. Chauvière, L. Preziosi and C. Verdier, Cell Mechanics: From single scale-based models to multiscale modeling, Journal of Biological Dynamics, 8 (2014), 74-78. [9] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513. [10] J. Q. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier, Amsterdam, 2014. [11] S. N. Ethier and T. G. Kurtz, Markov Processes. Characterization and Convergence, John Wiley & Sons, Inc., New York, 1986. doi: 10.1002/9780470316658. [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. [13] S. Gailus and K. Spiliopoulos, Discrete-time statistical inference for multiscale diffusions, Multiscale Model. Simul., 16 (2018), 1824-1858.  doi: 10.1137/17M1147408. [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. [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. [16] W. Janke, Rugged Free Energy Landscapes: Common Computational Approaches to Spin Glasses, Structural Glasses and Biological Macromolecules, Springer-Verlag Berlin Heidelberg, 2008. [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. [18] R. Z. Khasminskii, On the principle of averaging the Itô's stochastic differential equations, Kybernetika (Prague), 4 (1968), 260-279. [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. [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. [21] W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4. [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. [23] A. J. Majda, C. 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. [24] A. J. Majda, I. 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. [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. [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. [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. [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. [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. [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. [31] G. A. Pavliotis and A. M. Stuart, Multiscale Methods. Averaging and Homogenization, Springer, New York, 2008. [32] S. Peszat and J. Zabczyk, Strong Feller property and irreducibility for diffusions on Hilbert spaces, Ann. Probab., 23 (1995), 157-172. [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. [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.
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