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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

[2]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080

[3]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[4]

Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465

[5]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[6]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076

[7]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121

[8]

Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020071

[9]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018

[10]

Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168

[11]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[12]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052

[13]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[14]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[15]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[16]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[17]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241

[18]

Yahia Zare Mehrjerdi. A new methodology for solving bi-criterion fractional stochastic programming. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020054

[19]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[20]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (70)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]