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Probabilistic continuity of a pullback random attractor in time-sample

 1 Faculty of Eduction, Southwest University, Chongqing 400715, China 2 School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: Yangrong Li

Received  March 2019 Revised  August 2019 Published  February 2020

Fund Project: Li was supported by Natural Science Foundation of China grant 11571283, Wang was supported by National Social Science Foundation of China, 13th Five-year plan of Education grant BJA160059

Given a time-sample dependent attractor of a random dynamical system, we study its lower semi-continuity in probability along the time axis, and the criteria are established by using the local-sample asymptotically compactness for a triple-continuous system. The abstract results are applied to the non-autonomous stochastic p-Laplace equation on an unbounded domain with weakly dissipative nonlinearity. Without any additional hypotheses, we prove that the pullback random attractor is probabilistically continuous in both time and sample parameters.

Citation: Shulin Wang, Yangrong Li. Probabilistic continuity of a pullback random attractor in time-sample. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020028
References:
 [1] L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar [2] J. M. Arrieta, A. N. Carvalho and J. A. Langa, Continuity of dynamical structures for nonautonomous evolution equations under singular perturbations, J. Dynam. Differential Equations, 24 (2012), 427-481.  doi: 10.1007/s10884-012-9269-y.  Google Scholar [3] P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar [4] Z. Brzezniak and Y. Li, Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains, Tran. Amer. Math. Soc., 358 (2006), 5587-5629.  doi: 10.1090/S0002-9947-06-03923-7.  Google Scholar [5] T. Caraballo, A. N. Carvalho and H. B. Da Costa, Equi-attraction and continuity of attractors for skew-product semiflows, Discrete Contin. Dyn. Syst. Ser. 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Langa, Squeezing and finite dimensionality of cocycle attractors for 2D stochastic Navier-Stokes equation with non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1297-1324.  doi: 10.3934/dcdsb.2018152.  Google Scholar [11] H. Cui, M. M. Freitas and J. A. Langa, On random cocycle attractors with autonomous attraction universes, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3379-3407.  doi: 10.3934/dcdsb.2017142.  Google Scholar [12] H. Cui and J. A. Langa, Uniform attractors for non-autonomous random dynamical systems, J. Differential Equations, 263 (2017), 1225-1268.  doi: 10.1016/j.jde.2017.03.018.  Google Scholar [13] H. Cui, J. A. Langa and Y. Li, Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynam. Differential Equations, 30 (2018), 1873-1898.  doi: 10.1007/s10884-017-9617-z.  Google Scholar [14] H. Cui, P. E. Kloeden and F. Wu, Pathwise upper semi-continuity of random pullback attractors along the time axis, Physica D, 374 (2018), 21-34.  doi: 10.1016/j.physd.2018.03.002.  Google Scholar [15] R. N. Figueroa-Lopez and G. Lozada-Cruz, Dynamics of parabolic equations via the finite element method I. Continuity of the set of equilibria, J. Differential Equations, 261 (2016), 5235-5259.  doi: 10.1016/j.jde.2016.07.023.  Google Scholar [16] M. M. Freitas, P. Kalita and J. A. Langa, Continuity of non-autonomous attractors for hyperbolic perturbation of parabolic equations, J. Differential Equations, 264 (2018), 1886-1945.  doi: 10.1016/j.jde.2017.10.007.  Google Scholar [17] B. Gess, Random attractors for singular stochastic evolution equations, J. Differential Equations, 255 (2013), 524-559.  doi: 10.1016/j.jde.2013.04.023.  Google Scholar [18] B. Gess, Random attractors for degenerate stochastic partial differential equations, J. Dynam. Differential Equations, 25 (2013), 121-157.  doi: 10.1007/s10884-013-9294-5.  Google Scholar [19] A. Gu and P. E. Kloeden, Asymptotic behavior of a nonautonomous $p$-Laplacian lattice system, Intern. J. Bifur. Chaos, 26 (2016), 1650174, 9pp. doi: 10.1142/S0218127416501741.  Google Scholar [20] X. Han and P. E. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261 (2016), 2986-3009.  doi: 10.1016/j.jde.2016.05.015.  Google Scholar [21] L. T. Hoang, E. J. Olson and J. C. Robinson, Continuity of pullback and uniform attractors, J. Differential Equations, 264 (2018), 4067-4093.  doi: 10.1016/j.jde.2017.12.002.  Google Scholar [22] P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond., 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.  Google Scholar [23] P. E. Kloeden and J. Simsen, Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents, J. Math. Anal. Appl., 425 (2015), 911-918.  doi: 10.1016/j.jmaa.2014.12.069.  Google Scholar [24] P. E. Kloeden, J. Simsen and M. S. Simsen, Asymptotically autonomous multivalued cauchy problems with spatially variable exponents, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004.  Google Scholar [25] A. Krause, M. Lewis and B. Wang, Dynamics of the non-autonomous stochastic p-Laplace equation driven by multiplicative noise, Appl. Math. Comput., 246 (2014), 365-376.  doi: 10.1016/j.amc.2014.08.033.  Google Scholar [26] A. Krause and B. Wang, Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains, J. Math. Anal. Appl., 417 (2014), 1018-1038.  doi: 10.1016/j.jmaa.2014.03.037.  Google Scholar [27] J. A. Langa, J. C. Robinson and A. Suarez, The stability of attractors for non-autonomous perturbations of gradient-like systems, J. Differential Equations, 234 (2007), 607-625.  doi: 10.1016/j.jde.2006.11.016.  Google Scholar [28] D. Li, K. Lu, B. Wang and X. Wang, Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains, Discrete Contin. Dyn. Syst., 38 (2018), 187-208.  doi: 10.3934/dcds.2018009.  Google Scholar [29] F. Li, Y. Li and R. Wang, Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise, Discrete Cont. Dyn. Syst., 38 (2018), 3663-3685.  doi: 10.3934/dcds.2018158.  Google Scholar [30] Y. Li, A. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.  Google Scholar [31] Y. Li and B. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differential Equations, 245 (2008), 1775-1800.  doi: 10.1016/j.jde.2008.06.031.  Google Scholar [32] Y. Li, L. She and R. Wang, Asymptotically autonomous dynamics for parabolic equations, J. Math. Anal. Appl., 459 (2018), 1106-1123.  doi: 10.1016/j.jmaa.2017.11.033.  Google Scholar [33] Y. Li, L. She and J. Yin, Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1535-1557.  doi: 10.3934/dcdsb.2018058.  Google Scholar [34] Y. Li, R. Wang and J. Yin, Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2569-2586.  doi: 10.3934/dcdsb.2017092.  Google Scholar [35] Y. Li and J. Yin, A modified proof of pullback attractors in a Sobolev space for stochastic Fitzhugh-Nagumo equations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1203-1223.  doi: 10.3934/dcdsb.2016.21.1203.  Google Scholar [36] L. Liu and T. Caraballo, Well-posedness and dynamics of a fractional stochastic integro-differential equation, Physica D, 455 (2017), 45-57.  doi: 10.1016/j.physd.2017.05.006.  Google Scholar [37] L. Liu and X. Fu, Existence and upper semicontinuity of (L-2, L-q) pullback attractors for a stochastic p-Laplacian equation, Commun. Pure Appl. Anal., 16 (2017), 443-473.  doi: 10.3934/cpaa.2017023.  Google Scholar [38] J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Texts in Appl. Math. Cambridge, 2001.  Google Scholar [39] B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.  Google Scholar [40] B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.  doi: 10.1090/S0002-9947-2011-05247-5.  Google Scholar [41] S. Wang and Y. Li, Longtime robustness of pullback random attractors for stochastic magneto-hydrodynamics equations, Physica D, 382 (2018), 46-57.  doi: 10.1016/j.physd.2018.07.003.  Google Scholar [42] X. Wang, K. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.  Google Scholar [43] J. Yin and Y. Li, Two types of upper semi-continuity of bi-spatial attractors for non-autonomous stochastic p-Laplacian equations on R-n, Math. Methods Appl. Sci., 40 (2017), 4863-4879.   Google Scholar [44] W. Zhao, Random dynamics of stochastic p-Laplacian equations on R-N with an unbounded additive noise, J. Math. Anal. Appl., 455 (2017), 1178-1203.  doi: 10.1016/j.jmaa.2017.06.025.  Google Scholar [45] S. Zhou, Random exponential attractor for stochastic reaction-diffusion equation with multiplicative noise in $R^3$, J. Differential Equations, 263 (2017), 6347-6383.  doi: 10.1016/j.jde.2017.07.013.  Google Scholar

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References:
 [1] L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar [2] J. M. Arrieta, A. N. Carvalho and J. A. Langa, Continuity of dynamical structures for nonautonomous evolution equations under singular perturbations, J. Dynam. Differential Equations, 24 (2012), 427-481.  doi: 10.1007/s10884-012-9269-y.  Google Scholar [3] P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar [4] Z. Brzezniak and Y. Li, Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains, Tran. Amer. Math. Soc., 358 (2006), 5587-5629.  doi: 10.1090/S0002-9947-06-03923-7.  Google Scholar [5] T. Caraballo, A. N. Carvalho and H. B. Da Costa, Equi-attraction and continuity of attractors for skew-product semiflows, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2949-2967.  doi: 10.3934/dcdsb.2016081.  Google Scholar [6] M. D. Chekroun, E. Simonnet and M. Ghil, Stochastic climate dynamics: Random attractors and time-dependent invariant measures, Physica D, 240 (2011), 1685-1700.  doi: 10.1016/j.physd.2011.06.005.  Google Scholar [7] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar [8] H. Crauel, P. E. Kloeden and M. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains, Stoch. Dyn., 11 (2011), 301-314.  doi: 10.1142/S0219493711003292.  Google Scholar [9] H. Crauel and M. Scheutzow, Minimal random attractors, J. Differential Equations, 265 (2018), 702-718.  doi: 10.1016/j.jde.2018.03.011.  Google Scholar [10] H. Cui, M. M. Freitas and J. A. Langa, Squeezing and finite dimensionality of cocycle attractors for 2D stochastic Navier-Stokes equation with non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1297-1324.  doi: 10.3934/dcdsb.2018152.  Google Scholar [11] H. Cui, M. M. Freitas and J. A. Langa, On random cocycle attractors with autonomous attraction universes, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3379-3407.  doi: 10.3934/dcdsb.2017142.  Google Scholar [12] H. Cui and J. A. Langa, Uniform attractors for non-autonomous random dynamical systems, J. Differential Equations, 263 (2017), 1225-1268.  doi: 10.1016/j.jde.2017.03.018.  Google Scholar [13] H. Cui, J. A. Langa and Y. Li, Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynam. Differential Equations, 30 (2018), 1873-1898.  doi: 10.1007/s10884-017-9617-z.  Google Scholar [14] H. Cui, P. E. Kloeden and F. Wu, Pathwise upper semi-continuity of random pullback attractors along the time axis, Physica D, 374 (2018), 21-34.  doi: 10.1016/j.physd.2018.03.002.  Google Scholar [15] R. N. Figueroa-Lopez and G. Lozada-Cruz, Dynamics of parabolic equations via the finite element method I. Continuity of the set of equilibria, J. Differential Equations, 261 (2016), 5235-5259.  doi: 10.1016/j.jde.2016.07.023.  Google Scholar [16] M. M. Freitas, P. Kalita and J. A. Langa, Continuity of non-autonomous attractors for hyperbolic perturbation of parabolic equations, J. Differential Equations, 264 (2018), 1886-1945.  doi: 10.1016/j.jde.2017.10.007.  Google Scholar [17] B. Gess, Random attractors for singular stochastic evolution equations, J. Differential Equations, 255 (2013), 524-559.  doi: 10.1016/j.jde.2013.04.023.  Google Scholar [18] B. Gess, Random attractors for degenerate stochastic partial differential equations, J. Dynam. Differential Equations, 25 (2013), 121-157.  doi: 10.1007/s10884-013-9294-5.  Google Scholar [19] A. Gu and P. E. Kloeden, Asymptotic behavior of a nonautonomous $p$-Laplacian lattice system, Intern. J. Bifur. Chaos, 26 (2016), 1650174, 9pp. doi: 10.1142/S0218127416501741.  Google Scholar [20] X. Han and P. E. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261 (2016), 2986-3009.  doi: 10.1016/j.jde.2016.05.015.  Google Scholar [21] L. T. Hoang, E. J. Olson and J. C. Robinson, Continuity of pullback and uniform attractors, J. Differential Equations, 264 (2018), 4067-4093.  doi: 10.1016/j.jde.2017.12.002.  Google Scholar [22] P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond., 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.  Google Scholar [23] P. E. Kloeden and J. Simsen, Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents, J. Math. Anal. Appl., 425 (2015), 911-918.  doi: 10.1016/j.jmaa.2014.12.069.  Google Scholar [24] P. E. Kloeden, J. Simsen and M. S. Simsen, Asymptotically autonomous multivalued cauchy problems with spatially variable exponents, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004.  Google Scholar [25] A. Krause, M. Lewis and B. Wang, Dynamics of the non-autonomous stochastic p-Laplace equation driven by multiplicative noise, Appl. Math. Comput., 246 (2014), 365-376.  doi: 10.1016/j.amc.2014.08.033.  Google Scholar [26] A. Krause and B. Wang, Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains, J. Math. Anal. Appl., 417 (2014), 1018-1038.  doi: 10.1016/j.jmaa.2014.03.037.  Google Scholar [27] J. A. Langa, J. C. Robinson and A. Suarez, The stability of attractors for non-autonomous perturbations of gradient-like systems, J. Differential Equations, 234 (2007), 607-625.  doi: 10.1016/j.jde.2006.11.016.  Google Scholar [28] D. Li, K. Lu, B. Wang and X. Wang, Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains, Discrete Contin. Dyn. Syst., 38 (2018), 187-208.  doi: 10.3934/dcds.2018009.  Google Scholar [29] F. Li, Y. Li and R. Wang, Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise, Discrete Cont. Dyn. Syst., 38 (2018), 3663-3685.  doi: 10.3934/dcds.2018158.  Google Scholar [30] Y. Li, A. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.  Google Scholar [31] Y. Li and B. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differential Equations, 245 (2008), 1775-1800.  doi: 10.1016/j.jde.2008.06.031.  Google Scholar [32] Y. Li, L. She and R. Wang, Asymptotically autonomous dynamics for parabolic equations, J. Math. Anal. Appl., 459 (2018), 1106-1123.  doi: 10.1016/j.jmaa.2017.11.033.  Google Scholar [33] Y. Li, L. She and J. Yin, Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1535-1557.  doi: 10.3934/dcdsb.2018058.  Google Scholar [34] Y. Li, R. Wang and J. Yin, Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2569-2586.  doi: 10.3934/dcdsb.2017092.  Google Scholar [35] Y. Li and J. Yin, A modified proof of pullback attractors in a Sobolev space for stochastic Fitzhugh-Nagumo equations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1203-1223.  doi: 10.3934/dcdsb.2016.21.1203.  Google Scholar [36] L. Liu and T. Caraballo, Well-posedness and dynamics of a fractional stochastic integro-differential equation, Physica D, 455 (2017), 45-57.  doi: 10.1016/j.physd.2017.05.006.  Google Scholar [37] L. Liu and X. Fu, Existence and upper semicontinuity of (L-2, L-q) pullback attractors for a stochastic p-Laplacian equation, Commun. Pure Appl. Anal., 16 (2017), 443-473.  doi: 10.3934/cpaa.2017023.  Google Scholar [38] J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Texts in Appl. Math. Cambridge, 2001.  Google Scholar [39] B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.  Google Scholar [40] B. 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