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Two-time-scales hyperbolic-parabolic equations driven by Poisson random measures: existence, uniqueness and averaging principles. (English) Zbl 1387.60102
Summary: In this article, we are concerned with averaging principle for stochastic hyperbolic-parabolic equations driven by Poisson random measures with slow and fast time-scales. We first establish the existence and uniqueness of weak solutions of the stochastic hyperbolic-parabolic equations. Then, under suitable conditions, we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the stochastic wave equation is an average with respect to the stationary measure of the fast varying process. Finally, we derive the rate of strong convergence for the slow component towards the solution of the averaged equation.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
35M33 Initial-boundary value problems for mixed-type systems of PDEs
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[1] Applebaum, D., Lévy processes and stochastic calculus, (2009), Cambridge University Press Cambridge · Zbl 1200.60001
[2] Bertoin, J., Lévy processes, (1998), Cambridge University Press Cambridge · Zbl 0938.60005
[3] Bo, L.; Shi, K.; Wang, Y., On a stochastic wave equation driven by a non-Gaussian Lévy process, J. Theoret. Probab., 23, 1, 328-343, (2010) · Zbl 1196.60113
[4] Bréhier, C., Strong and weak orders in averaging for SPDEs, Stochastic Process. Appl., 122, 7, 2553-2593, (2012) · Zbl 1266.60112
[5] Cardetti, F.; Choi, Y. S., A parabolic-hyperbolic system modelling a moving cell, Electron. J. Differential Equations, 95, 1-11, (2009) · Zbl 1178.35389
[6] Cerrai, S., A khasminskii type averaging principle for stochastic reaction-diffusion equations, Ann. Appl. Probab., 19, 3, 899-948, (2009) · Zbl 1191.60076
[7] Cerrai, S.; Freidlin, M., Averaging principle for a class of stochastic reaction-diffusion equations, Probab. Theory Related Fields, 144, 1-2, 137-177, (2009) · Zbl 1176.60049
[8] Choi, Y.; Miller, C., Global existence of solutions to a coupled parabolic-hyperbolic system with moving boundary, Proc. Amer. Math. Soc., 139, 9, 3257-3270, (2011) · Zbl 1232.35197
[9] Chow, P., Thermoelastic wave propagation in a random medium and some related problems, Internat. J. Engrg. Sci., 11, 9, 953-971, (1973) · Zbl 0263.73007
[10] Chow, P., Stochastic partial differential equations, (2014), CRC Press
[11] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (2014), Cambridge University Press Cambridge · Zbl 1317.60077
[12] Duan, J., An introduction to stochastic dynamics, (2015), Cambridge University Press Cambridge · Zbl 1359.60003
[13] Duan, J.; Wang, W., Effective dynamics of stochastic partial differential equations, (2014), Elsevier · Zbl 1298.60006
[14] Freidlin, M.; Wentzell, A., Random perturbations of dynamical systems, (2012), Springer Science & Business Media Berlin, Heidelberg
[15] Fu, H.; Liu, J., Strong convergence in stochastic averaging principle for two time-scales stochastic partial differential equations, J. Math. Anal. Appl., 384, 1, 70-86, (2011) · Zbl 1223.60044
[16] Fu, H.; Liu, J.; Wan, L., Hyperbolic type stochastic evolution equations with Lévy noise, Acta Appl. Math., 125, 1, 193-208, (2013) · Zbl 1276.60065
[17] Fu, H.; Wan, L.; Wang, Y.; Liu, J., Strong convergence rate in averaging principle for stochastic Fitzhugh-Nagumo system with two time-scales, J. Math. Anal. Appl., 416, 2, 609-628, (2014) · Zbl 1325.60107
[18] Fu, H.; Wan, L.; Liu, J., Strong convergence in averaging principle for stochastic hyperbolic-parabolic equations with two time-scales, Stochastic Process. Appl., 125, 8, 3255-3279, (2015) · Zbl 1322.60111
[19] Givon, D., Strong convergence rate for two-time-scale jump-diffusion stochastic differential systems, Multiscale Model. Simul., 6, 2, 577-594, (2007) · Zbl 1144.60038
[20] Khasminskii, R., A limit theorem for the solutions of differential equations with random right-hand sides, Theory Probab. Appl., 11, 3, 390-406, (1966)
[21] Leung, A., Asymptotically stable invariant manifold for coupled nonlinear parabolic-hyperbolic partial differential equations, J. Differential Equations, 187, 184-200, (2003) · Zbl 1022.35006
[22] Michael, R.; Zhang, T., Stochastic evolution equations of jump type: existence, uniqueness and large deviation principles, Potential Anal., 26, 3, 255-279, (2007) · Zbl 1119.60057
[23] Mueller, C., The heat equation with Lévy noise, Stochastic Process. Appl., 74, 67-82, (1998) · Zbl 0934.60056
[24] Øksendal, B., Stochastic differential equations, (2003), Springer Berlin Heidelberg
[25] Peszat, S.; Zabczyk, J., Stochastic partial differential equations with Lévy noise: an evolution equation approach, (2007), Cambridge University Press Cambridge · Zbl 1205.60122
[26] Thompson, W.; Kuske, R.; Monahan, A., Stochastic averaging of dynamical systems with multiple time scales forced with α-stable noise, Multiscale Model. Simul., 13, 4, 1194-1223, (2015) · Zbl 1333.34099
[27] Wu, S.; Chen, H.; Li, W., The local and global existence of the solutions of hyperbolic-parabolic system modeling biological phenomena, Acta Math. Sci., 28, 1, 101-116, (2008) · Zbl 1150.35016
[28] Xu, Y.; Duan, J.; Xu, W., An averaging principle for stochastic dynamical systems with Lévy noise, Phys. D, 240, 17, 1395-1401, (2011) · Zbl 1236.60060
[29] Xu, Y.; Guo, R.; Liu, D.; Zhang, H.; Duan, J., Stochastic averaging principle for dynamical systems with fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 19, 4, 1197-1212, (2014) · Zbl 1314.60122
[30] Xu, J.; Miao, Y.; Liu, J., Strong averaging principle for slow-fast SPDEs with Poisson random measures, Discrete Contin. Dyn. Syst. Ser. B, 20, 7, 2233-2256, (2015) · Zbl 1335.60118
[31] Xu, Y.; Pei, B.; Guo, R., Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 20, 7, 2257-2267, (2015) · Zbl 1335.34090
[32] Xu, Y.; Pei, B.; Li, Y., Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise, Math. Methods Appl. Sci., 38, 11, 2120-2131, (2015) · Zbl 1345.60051
[33] Xu, Y.; Pei, B.; Wu, J.-L., Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion, Stoch. Dyn., 17, 2, (2017) · Zbl 1365.34102
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