September  2015, 20(7): 2233-2256. doi: 10.3934/dcdsb.2015.20.2233

Strong averaging principle for slow-fast SPDEs with Poisson random measures

1. 

College of Mathematics and Information Science, and Henan Engineering, Laboratory for Big Data Statistical Analysis and Optimal Control, Henan Normal University, Xinxiang, Henan 453007, China, China

2. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

Received  July 2014 Revised  December 2014 Published  July 2015

This work concerns the problem associated with an averaging principle for two-time-scales stochastic partial differential equations (SPDEs) driven by cylindrical Wiener processes and Poisson random measures. Under suitable dissipativity conditions, the existence of an averaging equation eliminating the fast variable for the coupled system is proved, and as a consequence, the system can be reduced to a single SPDE with a modified coefficient. Moreover, it is shown that the slow component mean-square strongly converges to the solution of the corresponding averaging equation.
Citation: Jie Xu, Yu Miao, Jicheng Liu. Strong averaging principle for slow-fast SPDEs with Poisson random measures. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2233-2256. doi: 10.3934/dcdsb.2015.20.2233
References:
[1]

J. Bao, A. Truman and C. Yuan, Stability in distribution of mild solutions to stochastic partial differential delay equations with jump,, Proc. R. Soc Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 2111. doi: 10.1098/rspa.2008.0486.

[2]

J. Bao, A. Truman and C. Yuan, Almost sure asymptotic Stability of stochastic partial differential equations with jump,, SIAM J. Control Optim., 49 (2011), 771. doi: 10.1137/100786812.

[3]

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

[4]

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

[5]

G. Cao and K. He, Successive approximations of infinite dimensional semilinear backward stochastic evolutions with jump,, Stoch. Proc. Appl., 117 (2007), 1251. doi: 10.1016/j.spa.2007.01.003.

[6]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223.

[7]

A. Debussche, M. Hogele and P. Imkeller, The dynamics of nonlinear reaction-diffusion equations with small Lévy noise,, Springer Lecture notes in Mathematics, (2013).

[8]

Z. Dong and T. Xu, One-dimensional stochastic Burgers equation driven by Lévy processes,, J. Funct. Anal., 243 (2007), 631. doi: 10.1016/j.jfa.2006.09.010.

[9]

D. Filipovic, S. Tappe and J. Teichmann, Jump-difusions in Hilbert spaces: Existence, stability and numerics,, Stochastics, 82 (2010), 475. doi: 10.1080/17442501003624407.

[10]

H. Fu and J. Liu, Strong convergence in stochastic averaging for two time-scales stochastic partial differential equations,, J. Math. Anal. Appl., 384 (2011), 70. doi: 10.1016/j.jmaa.2011.02.076.

[11]

H. Fu and J. Duan, An averaging principle for two time-scales stochastic partial differential equations,, Stochastic and Dynamics, 11 (2011), 353. doi: 10.1142/S0219493711003346.

[12]

D. Givon, Strong convergence rate for two-time-scale jump-diffusion stochastic differential sysytems,, SIAM J. Multi. Model. Simul., 6 (2007), 577. doi: 10.1137/060673345.

[13]

J. Golec, Stochastic averaging principle for systems with pathwise uniqueness,, Stochastic Anal. Appl., 13 (1995), 307. doi: 10.1080/07362999508809400.

[14]

E. Hausenblas, Existence, uniqueness and regularity of SPDEs driven by Poisson random measures,, Electron. J. Probab., 10 (2005), 1496. doi: 10.1214/EJP.v10-297.

[15]

W. Jia and W. Zhu, Stochastic averaging of quasi-partially integrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations,, Physica A: Statistical Mechanics and its Applications, 398 (2014), 125. doi: 10.1016/j.physa.2013.12.009.

[16]

W. Jia, W. Zhu and Y. Xu, Stochastic averaging of quasi-non-integrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations,, International Journal of Non-Linear Mechanics, 51 (2013), 45.

[17]

R. Z. Khasminskii, On the principle of averaging the Itô stochastic differential equations (in Russian),, kibernetika., 4 (1968), 260.

[18]

D. Liu, Strong convergence of principle of averaging for multiscale stochastic dynamical systems,, Commun. Math. Sci., 8 (2010), 999. doi: 10.4310/CMS.2010.v8.n4.a11.

[19]

W. Liu and M. Stephan, Yosida approximations for multivalued stochastic partial differential equations driven by Lévy noise on a Gelfand triple,, J. Math. Anal. Appl., 410 (2014), 158. doi: 10.1016/j.jmaa.2013.08.016.

[20]

A. Løkka, B. Øksendal and F. Proske, Stochastic partial differential equations driven by Lévy space-time white noise,, The Annals of Applied Probability, 14 (2004), 1506. doi: 10.1214/105051604000000413.

[21]

B. Øksendal, Stochastic Differential Equations,, $6^{th}$ edition, (2003). doi: 10.1007/978-3-642-14394-6.

[22]

A. Pazy, Semigroups of Linear Operations and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1.

[23]

S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise,, Encyclopedia Mathematics, (2007). doi: 10.1017/CBO9780511721373.

[24]

M. Röckner and T. Zhang, Stochastic evolution equation of jump type: Existence, uniqueness and large deviation principles,, Potential Anal., 26 (2007), 255. doi: 10.1007/s11118-006-9035-z.

[25]

J. Seidler and I. Vrkoč, An averaging principle for stochastic evolution equations,, Časopis Pěst. Mat., 115 (1990), 240.

[26]

T. Taniguchi, The existence and asymptotic behaviour of energy solutions to stochastic 2D functional Navier-Stokes equations driven by Lévy processes,, J. Math. Anal. Appl., 385 (2012), 634. doi: 10.1016/j.jmaa.2011.06.076.

[27]

W. Wang and A. J. Roberts, Average and deviation for slow-fast SPDEs,, J. Differential Equations, 253 (2012), 1265. doi: 10.1016/j.jde.2012.05.011.

[28]

G. Yin and H. Yang, Two-time-scale jump-diffusion models with Markovian switching regimes,, Stochastics and Stochastic Reports, 76 (2004), 77. doi: 10.1080/10451120410001696261.

show all references

References:
[1]

J. Bao, A. Truman and C. Yuan, Stability in distribution of mild solutions to stochastic partial differential delay equations with jump,, Proc. R. Soc Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 2111. doi: 10.1098/rspa.2008.0486.

[2]

J. Bao, A. Truman and C. Yuan, Almost sure asymptotic Stability of stochastic partial differential equations with jump,, SIAM J. Control Optim., 49 (2011), 771. doi: 10.1137/100786812.

[3]

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

[4]

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

[5]

G. Cao and K. He, Successive approximations of infinite dimensional semilinear backward stochastic evolutions with jump,, Stoch. Proc. Appl., 117 (2007), 1251. doi: 10.1016/j.spa.2007.01.003.

[6]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223.

[7]

A. Debussche, M. Hogele and P. Imkeller, The dynamics of nonlinear reaction-diffusion equations with small Lévy noise,, Springer Lecture notes in Mathematics, (2013).

[8]

Z. Dong and T. Xu, One-dimensional stochastic Burgers equation driven by Lévy processes,, J. Funct. Anal., 243 (2007), 631. doi: 10.1016/j.jfa.2006.09.010.

[9]

D. Filipovic, S. Tappe and J. Teichmann, Jump-difusions in Hilbert spaces: Existence, stability and numerics,, Stochastics, 82 (2010), 475. doi: 10.1080/17442501003624407.

[10]

H. Fu and J. Liu, Strong convergence in stochastic averaging for two time-scales stochastic partial differential equations,, J. Math. Anal. Appl., 384 (2011), 70. doi: 10.1016/j.jmaa.2011.02.076.

[11]

H. Fu and J. Duan, An averaging principle for two time-scales stochastic partial differential equations,, Stochastic and Dynamics, 11 (2011), 353. doi: 10.1142/S0219493711003346.

[12]

D. Givon, Strong convergence rate for two-time-scale jump-diffusion stochastic differential sysytems,, SIAM J. Multi. Model. Simul., 6 (2007), 577. doi: 10.1137/060673345.

[13]

J. Golec, Stochastic averaging principle for systems with pathwise uniqueness,, Stochastic Anal. Appl., 13 (1995), 307. doi: 10.1080/07362999508809400.

[14]

E. Hausenblas, Existence, uniqueness and regularity of SPDEs driven by Poisson random measures,, Electron. J. Probab., 10 (2005), 1496. doi: 10.1214/EJP.v10-297.

[15]

W. Jia and W. Zhu, Stochastic averaging of quasi-partially integrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations,, Physica A: Statistical Mechanics and its Applications, 398 (2014), 125. doi: 10.1016/j.physa.2013.12.009.

[16]

W. Jia, W. Zhu and Y. Xu, Stochastic averaging of quasi-non-integrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations,, International Journal of Non-Linear Mechanics, 51 (2013), 45.

[17]

R. Z. Khasminskii, On the principle of averaging the Itô stochastic differential equations (in Russian),, kibernetika., 4 (1968), 260.

[18]

D. Liu, Strong convergence of principle of averaging for multiscale stochastic dynamical systems,, Commun. Math. Sci., 8 (2010), 999. doi: 10.4310/CMS.2010.v8.n4.a11.

[19]

W. Liu and M. Stephan, Yosida approximations for multivalued stochastic partial differential equations driven by Lévy noise on a Gelfand triple,, J. Math. Anal. Appl., 410 (2014), 158. doi: 10.1016/j.jmaa.2013.08.016.

[20]

A. Løkka, B. Øksendal and F. Proske, Stochastic partial differential equations driven by Lévy space-time white noise,, The Annals of Applied Probability, 14 (2004), 1506. doi: 10.1214/105051604000000413.

[21]

B. Øksendal, Stochastic Differential Equations,, $6^{th}$ edition, (2003). doi: 10.1007/978-3-642-14394-6.

[22]

A. Pazy, Semigroups of Linear Operations and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1.

[23]

S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise,, Encyclopedia Mathematics, (2007). doi: 10.1017/CBO9780511721373.

[24]

M. Röckner and T. Zhang, Stochastic evolution equation of jump type: Existence, uniqueness and large deviation principles,, Potential Anal., 26 (2007), 255. doi: 10.1007/s11118-006-9035-z.

[25]

J. Seidler and I. Vrkoč, An averaging principle for stochastic evolution equations,, Časopis Pěst. Mat., 115 (1990), 240.

[26]

T. Taniguchi, The existence and asymptotic behaviour of energy solutions to stochastic 2D functional Navier-Stokes equations driven by Lévy processes,, J. Math. Anal. Appl., 385 (2012), 634. doi: 10.1016/j.jmaa.2011.06.076.

[27]

W. Wang and A. J. Roberts, Average and deviation for slow-fast SPDEs,, J. Differential Equations, 253 (2012), 1265. doi: 10.1016/j.jde.2012.05.011.

[28]

G. Yin and H. Yang, Two-time-scale jump-diffusion models with Markovian switching regimes,, Stochastics and Stochastic Reports, 76 (2004), 77. doi: 10.1080/10451120410001696261.

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