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Weak time discretization for slow-fast stochastic reaction-diffusion equations
1. | Department of Mathematics, Nanjing University, Nanjing 210093, China |
2. | School of Information, Nanjing Audit University, 211815, China |
This paper derives a weak convergence theorem of the time discretization of the slow component for a two-time-scale stochastic evolutionary equations on interval [0, 1]. Here the drift coefficient of the slow component is cubic with linear coupling between slow and fast components.
References:
[1] |
J. Bao, G. Yin and C. Yuan,
Two-time-scale stochastic partial differential equations driven by $\alpha$-stable noises: averaging principles, Bernoulli., 23 (2017), 645-669.
doi: 10.3150/14-BEJ677. |
[2] |
R. Bertram and J. E. Rubin,
Multi-time scale systems and fast-slow analysis, Math. Biosci., 287 (2017), 105-121.
doi: 10.1016/j.mbs.2016.07.003. |
[3] |
C.-E. Bréhier,
Strong and weak orders in averaging for SPDEs, Stoch. Proc. Appl., 122 (2012), 2553-2593.
doi: 10.1016/j.spa.2012.04.007. |
[4] |
C.-E. Bréhier,
Approximation of the invariant measure with an Euler scheme for stochastic PDEs driven by space-time white noise, Potential Analysis., 40 (2014), 1-40.
doi: 10.1007/s11118-013-9338-9. |
[5] |
S. Cerrai,
A Khasminskii type averaging principle for stochastic reaction-diffusion equations, Ann. Appl. Prob., 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, Prob. Th. Related Fields, 144 (2009), 137-177.
doi: 10.1007/s00440-008-0144-z. |
[7] |
Z. Dong, X. Sun, H. Xiao and J. Zhai,
Averaging principle for one dimensional stochastic Burgers equation, J. Diff. Equa., 265 (2018), 4749-4797.
doi: 10.1016/j.jde.2018.06.020. |
[8] |
S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2003, Available from: https://www.researchgate.net/publication/228607920. |
[9] |
J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier Press, London, 2014.
doi: 10.1016/C2013-0-15235-X.![]() |
[10] |
W. E, Analysis of the heterogeneous multiscale method for ordinary differential equations, Comm. Math. Sci., 1 (2003), 423–436. https://projecteuclid.org/euclid.cms/1250880094. |
[11] |
W. E and B. Engquist, Multiscale modeling and computations, Notice of AMS, 50 (2003), 1062–1070. http://hdl.handle.net/1903/9877 Google Scholar |
[12] |
W. E and B. Engquist, The heterogeneous multiscale methods, Comm. Math. Sci., 1 (2003), 87-132. Google Scholar |
[13] |
W. E, D. Liu and E. Vanden-Eijnden,
Analysis of multiscale methods for stochastic differential equations, Comm. Pure Appl. Math., 58 (2005), 1544-1585.
doi: 10.1002/cpa.20088. |
[14] |
W. E, P. Ming and P. Zhang,
Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc., 18 (2005), 121-156.
doi: 10.1090/S0894-0347-04-00469-2. |
[15] |
H. Fu, L. Wan and J. Liu, Weak order in averaging principle for two-time-scale stochastic partial differential equations, preprint, arXiv: 1802.00903. Google Scholar |
[16] |
H. Fu, L. Wan, Y. Wang, J. Liu and X. Liu,
Strong convergence rate in averaging principle for stochastic FitzHug-Nagumo system with two time-scales, J. Math. Anal. Appl., 416 (2014), 609-628.
doi: 10.1016/j.jmaa.2014.02.062. |
[17] |
E. Harvey, V. Kirk, M. Wechselberger and J. Sneyd,
Multiple timescales, mixed mode oscillations and canards in models of intracellular calcium dynamics, J. Nonlinear Sci., 21 (2011), 639-683.
doi: 10.1007/s00332-011-9096-z. |
[18] |
R. Z. Khasminskiǐ,
On the averaging principle of the Itô stochastic differential equations, Kibernetika, 4 (1968), 260-279.
|
[19] |
N. Krylov and N. Bogliubov, Introduction to Nonlinear Mechanics, Princeton University Press, Princeton, 1943.
![]() |
[20] |
J. A. Sauders and F. Verhulst, Averaging Methods to Nonlinear Dynamical Systems, Springer–Verlag, New York, 1985.
doi: 10.1007/978-1-4757-4575-7. |
[21] |
N. G. Sorokina,
On a theorem of N.N. Bogoliubov, Ukrain. Mat. Zh., 4 (1959), 220-222.
|
[22] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer–Verlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[23] |
E. Vanden-Eijnden,
Numerical techniques for multiscale dynamical systems with stochastic effects, Comm. Math. Sci., 1 (2003), 385-391.
doi: 10.4310/CMS.2003.v1.n2.a11. |
[24] |
V. M. Volosov,
Averaging in systems of ordinary differential equations, Russian Math. Surveys, 17 (1962), 1-126.
|
[25] |
W. Wang and A. J. Roberts,
Average and deviation for slow-fast stochastic partial differential equations, J. Diff. Equa., 253 (2012), 1265-1286.
doi: 10.1016/j.jde.2012.05.011. |
[26] |
W. Wang and A. J. Roberts,
Slow manifold and averaging for slow-fast stochastic system, J. Math. Anal. Appl., 398 (2013), 822-839.
doi: 10.1016/j.jmaa.2012.09.029. |
[27] |
Y. Xu, J. Q. Duan and W. Xu,
An averaging principle for stochastic dynamical systems with Lévy noise, Physica D, 240 (2011), 1395-1401.
doi: 10.1016/j.physd.2011.06.001. |
show all references
References:
[1] |
J. Bao, G. Yin and C. Yuan,
Two-time-scale stochastic partial differential equations driven by $\alpha$-stable noises: averaging principles, Bernoulli., 23 (2017), 645-669.
doi: 10.3150/14-BEJ677. |
[2] |
R. Bertram and J. E. Rubin,
Multi-time scale systems and fast-slow analysis, Math. Biosci., 287 (2017), 105-121.
doi: 10.1016/j.mbs.2016.07.003. |
[3] |
C.-E. Bréhier,
Strong and weak orders in averaging for SPDEs, Stoch. Proc. Appl., 122 (2012), 2553-2593.
doi: 10.1016/j.spa.2012.04.007. |
[4] |
C.-E. Bréhier,
Approximation of the invariant measure with an Euler scheme for stochastic PDEs driven by space-time white noise, Potential Analysis., 40 (2014), 1-40.
doi: 10.1007/s11118-013-9338-9. |
[5] |
S. Cerrai,
A Khasminskii type averaging principle for stochastic reaction-diffusion equations, Ann. Appl. Prob., 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, Prob. Th. Related Fields, 144 (2009), 137-177.
doi: 10.1007/s00440-008-0144-z. |
[7] |
Z. Dong, X. Sun, H. Xiao and J. Zhai,
Averaging principle for one dimensional stochastic Burgers equation, J. Diff. Equa., 265 (2018), 4749-4797.
doi: 10.1016/j.jde.2018.06.020. |
[8] |
S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2003, Available from: https://www.researchgate.net/publication/228607920. |
[9] |
J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier Press, London, 2014.
doi: 10.1016/C2013-0-15235-X.![]() |
[10] |
W. E, Analysis of the heterogeneous multiscale method for ordinary differential equations, Comm. Math. Sci., 1 (2003), 423–436. https://projecteuclid.org/euclid.cms/1250880094. |
[11] |
W. E and B. Engquist, Multiscale modeling and computations, Notice of AMS, 50 (2003), 1062–1070. http://hdl.handle.net/1903/9877 Google Scholar |
[12] |
W. E and B. Engquist, The heterogeneous multiscale methods, Comm. Math. Sci., 1 (2003), 87-132. Google Scholar |
[13] |
W. E, D. Liu and E. Vanden-Eijnden,
Analysis of multiscale methods for stochastic differential equations, Comm. Pure Appl. Math., 58 (2005), 1544-1585.
doi: 10.1002/cpa.20088. |
[14] |
W. E, P. Ming and P. Zhang,
Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc., 18 (2005), 121-156.
doi: 10.1090/S0894-0347-04-00469-2. |
[15] |
H. Fu, L. Wan and J. Liu, Weak order in averaging principle for two-time-scale stochastic partial differential equations, preprint, arXiv: 1802.00903. Google Scholar |
[16] |
H. Fu, L. Wan, Y. Wang, J. Liu and X. Liu,
Strong convergence rate in averaging principle for stochastic FitzHug-Nagumo system with two time-scales, J. Math. Anal. Appl., 416 (2014), 609-628.
doi: 10.1016/j.jmaa.2014.02.062. |
[17] |
E. Harvey, V. Kirk, M. Wechselberger and J. Sneyd,
Multiple timescales, mixed mode oscillations and canards in models of intracellular calcium dynamics, J. Nonlinear Sci., 21 (2011), 639-683.
doi: 10.1007/s00332-011-9096-z. |
[18] |
R. Z. Khasminskiǐ,
On the averaging principle of the Itô stochastic differential equations, Kibernetika, 4 (1968), 260-279.
|
[19] |
N. Krylov and N. Bogliubov, Introduction to Nonlinear Mechanics, Princeton University Press, Princeton, 1943.
![]() |
[20] |
J. A. Sauders and F. Verhulst, Averaging Methods to Nonlinear Dynamical Systems, Springer–Verlag, New York, 1985.
doi: 10.1007/978-1-4757-4575-7. |
[21] |
N. G. Sorokina,
On a theorem of N.N. Bogoliubov, Ukrain. Mat. Zh., 4 (1959), 220-222.
|
[22] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer–Verlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[23] |
E. Vanden-Eijnden,
Numerical techniques for multiscale dynamical systems with stochastic effects, Comm. Math. Sci., 1 (2003), 385-391.
doi: 10.4310/CMS.2003.v1.n2.a11. |
[24] |
V. M. Volosov,
Averaging in systems of ordinary differential equations, Russian Math. Surveys, 17 (1962), 1-126.
|
[25] |
W. Wang and A. J. Roberts,
Average and deviation for slow-fast stochastic partial differential equations, J. Diff. Equa., 253 (2012), 1265-1286.
doi: 10.1016/j.jde.2012.05.011. |
[26] |
W. Wang and A. J. Roberts,
Slow manifold and averaging for slow-fast stochastic system, J. Math. Anal. Appl., 398 (2013), 822-839.
doi: 10.1016/j.jmaa.2012.09.029. |
[27] |
Y. Xu, J. Q. Duan and W. Xu,
An averaging principle for stochastic dynamical systems with Lévy noise, Physica D, 240 (2011), 1395-1401.
doi: 10.1016/j.physd.2011.06.001. |
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