April  2020, 25(4): 1565-1581. doi: 10.3934/dcdsb.2019240

Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations

School of Mathematical Science, and V.C. & V.R. Key Lab, Sichuan Normal University, Chengdu 610068, China

* Corresponding author: Guanggan Chen

Received  January 2019 Revised  April 2019 Published  April 2020 Early access  November 2019

This work provides a finite dimensional reducing and a smooth approximating for a class of stochastic partial differential equations with an additive white noise. Using the invariant random cone to show the asymptotical completion, this stochastic partial differential equation is reduced to a stochastic ordinary differential equation on a random invariant manifold. Furthermore, after deriving the finite dimensional reducing for another stochastic partial differential equation driven by a Wong-Zakai scheme via a smooth colored noise, it is proved that when the smooth colored noise tends to the white noise, the solution and the finite dimensional reducing of the approximate system converge pathwisely to those of the original system.

Citation: Min Yang, Guanggan Chen. Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1565-1581. doi: 10.3934/dcdsb.2019240
References:
[1]

P. Acquistapace and B. Terreni, An approach to Ito linear equations in Hilbert spaces by approximation of white noise with coloured noise, Stoch. Anal. Appl., 2 (1984), 131-186.  doi: 10.1080/07362998408809031.

[2]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[3]

D. Blömker, Amplitude Equations for Stochastic Partial Differential Equations, Interdisciplinary Mathematical Sciences, 3. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/6440.

[4]

D. Blömker and W. Wang, Qualitative properties of local random invariant manifolds for SPDE with quadratic nonlinearity, J. Dyn. Differ. Equ., 22 (2010), 677-695.  doi: 10.1007/s10884-009-9145-6.

[5]

T. CaraballoJ. Q. DuanK. N. Lu and B. Schmalfuss, Invariant manifolds for random and stochastic partial differential equations, Adv. Nonlinear Stud., 10 (2010), 23-52.  doi: 10.1515/ans-2010-0102.

[6]

G. G. Chen, J. Q. Duan and J. Zhang, Geometric shape of invariant manifolds for a class of stochastic partial differential equations, J. Math. Phys., 52 (2011), 072702. doi: 10.1063/1.3614777.

[7]

G. G. ChenJ. Q. Duan and J. Zhang, Approximating dynamics of a singularly perturbed stochastic wave equation with a random dynamical boundary condition, SIAM. J. Math. Anal., 45 (2013), 2790-2814.  doi: 10.1137/12088968X.

[8]

G. G. ChenJ. Q. Duan and J. Zhang, Slow foliation of a slow-fast stochastic evolutionary system, J. Funct. Anal., 267 (2014), 2663-2697.  doi: 10.1016/j.jfa.2014.07.031.

[9] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.
[10]

J. Q. DuanK. N. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.  doi: 10.1214/aop/1068646380.

[11]

J. Q. DuanK. N. Lu and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dyn. Differ. Equ., 16 (2004), 949-972.  doi: 10.1007/s10884-004-7830-z.

[12]

J. Q. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier Insights, Elsevier, Amsterdam, 2014.

[13]

J. García-Ojalvo and J. M. Sancho, Noise in Spatially Extended Systems, Institute for Nonlinear Science, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1536-3.

[14]

B. Gess, Random attractors for stochastic porous media equations perturbed by space-time linear multiplicative noise, Ann. Probab., 42 (2014), 818-864.  doi: 10.1214/13-AOP869.

[15]

Z. K. GuoX. J. YanW. F. Wang and X. M. Liu, Approximate the dynamical behavior for stochastic systems by Wong-Zakai approaching, J. Math. Anal. Appl., 457 (2018), 214-232.  doi: 10.1016/j.jmaa.2017.08.004.

[16]

M. Hairer and É. Pardoux, A Wong-Zakai theorem for stochastic PDEs, J. Math. Soc. Jpn., 67 (2015), 1551-1604.  doi: 10.2969/jmsj/06741551.

[17]

W. Horsthemke and R. Lefever, Noise-induced Transitions: Theory and Applications in Physics, Chemistry, and Biology, Springer Series in Synergetics, 15. Springer-Verlag, Berlin, 1984. doi: 10.1007/3-540-36852-3.

[18]

N. IkedaS. Nakao and Y. Yamato, A class of approximations of Brownian motion, Publ. Res. I. Math. Sci., 13 (1977), 285-300.  doi: 10.2977/prims/1195190109.

[19]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd edition, North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam, Kodansha, Ltd., Tokyo, 1989.

[20]

T. JiangX. M. Liu and J. Q. Duan, Approximation for random stable manifolds under multiplicative correlated noises, Discrete Continuous Dynam. Systems-B, 21 (2016), 3163-3174.  doi: 10.3934/dcdsb.2016091.

[21]

T. Jiang, X. M. Liu and J. Q. Duan, A Wong-Zakai approximation for random invariant manifolds, J. Math. Phys., 58 (2017), 122701. doi: 10.1063/1.5017932.

[22]

D. Kelley and I. Melbourne, Smooth approximation of stochastic differential equations, Ann. Probab., 44 (2016), 479-520.  doi: 10.1214/14-AOP979.

[23]

N. Koksch and S. Siegmund, Pullback attracting inertial manifolds for nonautonomous dynamical systems, J. Dyn. Differ. Equ., 14 (2002), 889-941.  doi: 10.1023/A:1020768711975.

[24]

F. Konecny, On Wong-Zakai approximation of stochastic differential equations, J. Multivariate Anal., 13 (1983), 605-611.  doi: 10.1016/0047-259X(83)90043-X.

[25]

T. G. Kurtz and P. Protter, Weak limit theorems for stochastic integrals and stochastic differential equations, Ann. Probab., 19 (1991), 1035-1070.  doi: 10.1214/aop/1176990334.

[26]

K. N. Lu and B. Schmalfuss, Invariant manifolds for stochastic wave equations, J. Differ. Equ., 236 (2007), 460-492.  doi: 10.1016/j.jde.2006.09.024.

[27]

E. J. McShane, Stochastic differential equations and models of random processes, Proc. Sixth Berkeley Symp. on Math. Statist. and Prob., 3 (1972), 263-294. 

[28]

S. A. MohammedT. S. Zhang and H. Z. Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Mem. Am. Math. Soc., 196 (2008), 1-105.  doi: 10.1090/memo/0917.

[29]

S. Nakao, On weak convergence of sequences of continuous local martingale, Ann. Inst. H. Poincaré Probab. Statist., 22 (1986), 371-380. 

[30]

P. Protter, Approximations of solutions of stochastic differential equations driven by semimartingales, Ann. Probab., 13 (1985), 716-743.  doi: 10.1214/aop/1176992905.

[31]

N. V. Krylov and B. L. Rozovski$\check{{\rm i}}$, Stochastic evolution equations, Current Problems in Mathematics, Vol. 14 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 256 (1979), 71–147.

[32]

J. Shen and K. N. Lu, Wong-Zakai approximations and center manifolds of stochastic differential equations, J. Differ. Equ., 263 (2017), 4929-4977.  doi: 10.1016/j.jde.2017.06.005.

[33]

J. ShenK. N. Lu and W. N. Zhang, Heteroclinic chaotic behavior driven by a Brownian motion, J. Differ. Equ., 255 (2013), 4185-4225.  doi: 10.1016/j.jde.2013.08.003.

[34]

D. W. Stook and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, Proc. Sixth Berkeley Symp. on Math. Statist. and Prob., 3 (1972), 333-359. 

[35]

H. J. Sussmann, An interpretation of stochastic differential equations as ordinary differential equations which depend on the sample point, Bull. Amer. Math. Soc., 83 (1977), 296-298.  doi: 10.1090/S0002-9904-1977-14312-7.

[36]

H. J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. Probab., 6 (1978), 19-41.  doi: 10.1214/aop/1176995608.

[37]

G. Tessitore and J. Zabczyk, Wong-Zakai approximation of stochastic evolution equations, J. Evol. Equ., 6 (2006), 621-655.  doi: 10.1007/s00028-006-0280-9.

[38]

X. H. WangK. N. Lu and B. X. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equ., 264 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.

[39]

W. Wang and J. Q. Duan, A dynamical approximation for stochastic partial differential equations, J. Math. Phys., 48 (2007), 102701. doi: 10.1063/1.2800164.

[40]

E. Wong and M. Zakai, On the relation between ordinary and stochastic differential equations, Int. J. Eng. Sci., 3 (1965), 213-229.  doi: 10.1016/0020-7225(65)90045-5.

[41]

E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Stat., 36 (1965), 1560-1564.  doi: 10.1214/aoms/1177699916.

[42]

X. T. YanX. M. Liu and M. H. Yang, Random attractors of stochastic partial differential equations: A smooth approximation approach, Stoch. Anal. Appl., 35 (2017), 1007-1029.  doi: 10.1080/07362994.2017.1345317.

show all references

References:
[1]

P. Acquistapace and B. Terreni, An approach to Ito linear equations in Hilbert spaces by approximation of white noise with coloured noise, Stoch. Anal. Appl., 2 (1984), 131-186.  doi: 10.1080/07362998408809031.

[2]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[3]

D. Blömker, Amplitude Equations for Stochastic Partial Differential Equations, Interdisciplinary Mathematical Sciences, 3. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/6440.

[4]

D. Blömker and W. Wang, Qualitative properties of local random invariant manifolds for SPDE with quadratic nonlinearity, J. Dyn. Differ. Equ., 22 (2010), 677-695.  doi: 10.1007/s10884-009-9145-6.

[5]

T. CaraballoJ. Q. DuanK. N. Lu and B. Schmalfuss, Invariant manifolds for random and stochastic partial differential equations, Adv. Nonlinear Stud., 10 (2010), 23-52.  doi: 10.1515/ans-2010-0102.

[6]

G. G. Chen, J. Q. Duan and J. Zhang, Geometric shape of invariant manifolds for a class of stochastic partial differential equations, J. Math. Phys., 52 (2011), 072702. doi: 10.1063/1.3614777.

[7]

G. G. ChenJ. Q. Duan and J. Zhang, Approximating dynamics of a singularly perturbed stochastic wave equation with a random dynamical boundary condition, SIAM. J. Math. Anal., 45 (2013), 2790-2814.  doi: 10.1137/12088968X.

[8]

G. G. ChenJ. Q. Duan and J. Zhang, Slow foliation of a slow-fast stochastic evolutionary system, J. Funct. Anal., 267 (2014), 2663-2697.  doi: 10.1016/j.jfa.2014.07.031.

[9] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.
[10]

J. Q. DuanK. N. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.  doi: 10.1214/aop/1068646380.

[11]

J. Q. DuanK. N. Lu and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dyn. Differ. Equ., 16 (2004), 949-972.  doi: 10.1007/s10884-004-7830-z.

[12]

J. Q. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier Insights, Elsevier, Amsterdam, 2014.

[13]

J. García-Ojalvo and J. M. Sancho, Noise in Spatially Extended Systems, Institute for Nonlinear Science, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1536-3.

[14]

B. Gess, Random attractors for stochastic porous media equations perturbed by space-time linear multiplicative noise, Ann. Probab., 42 (2014), 818-864.  doi: 10.1214/13-AOP869.

[15]

Z. K. GuoX. J. YanW. F. Wang and X. M. Liu, Approximate the dynamical behavior for stochastic systems by Wong-Zakai approaching, J. Math. Anal. Appl., 457 (2018), 214-232.  doi: 10.1016/j.jmaa.2017.08.004.

[16]

M. Hairer and É. Pardoux, A Wong-Zakai theorem for stochastic PDEs, J. Math. Soc. Jpn., 67 (2015), 1551-1604.  doi: 10.2969/jmsj/06741551.

[17]

W. Horsthemke and R. Lefever, Noise-induced Transitions: Theory and Applications in Physics, Chemistry, and Biology, Springer Series in Synergetics, 15. Springer-Verlag, Berlin, 1984. doi: 10.1007/3-540-36852-3.

[18]

N. IkedaS. Nakao and Y. Yamato, A class of approximations of Brownian motion, Publ. Res. I. Math. Sci., 13 (1977), 285-300.  doi: 10.2977/prims/1195190109.

[19]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd edition, North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam, Kodansha, Ltd., Tokyo, 1989.

[20]

T. JiangX. M. Liu and J. Q. Duan, Approximation for random stable manifolds under multiplicative correlated noises, Discrete Continuous Dynam. Systems-B, 21 (2016), 3163-3174.  doi: 10.3934/dcdsb.2016091.

[21]

T. Jiang, X. M. Liu and J. Q. Duan, A Wong-Zakai approximation for random invariant manifolds, J. Math. Phys., 58 (2017), 122701. doi: 10.1063/1.5017932.

[22]

D. Kelley and I. Melbourne, Smooth approximation of stochastic differential equations, Ann. Probab., 44 (2016), 479-520.  doi: 10.1214/14-AOP979.

[23]

N. Koksch and S. Siegmund, Pullback attracting inertial manifolds for nonautonomous dynamical systems, J. Dyn. Differ. Equ., 14 (2002), 889-941.  doi: 10.1023/A:1020768711975.

[24]

F. Konecny, On Wong-Zakai approximation of stochastic differential equations, J. Multivariate Anal., 13 (1983), 605-611.  doi: 10.1016/0047-259X(83)90043-X.

[25]

T. G. Kurtz and P. Protter, Weak limit theorems for stochastic integrals and stochastic differential equations, Ann. Probab., 19 (1991), 1035-1070.  doi: 10.1214/aop/1176990334.

[26]

K. N. Lu and B. Schmalfuss, Invariant manifolds for stochastic wave equations, J. Differ. Equ., 236 (2007), 460-492.  doi: 10.1016/j.jde.2006.09.024.

[27]

E. J. McShane, Stochastic differential equations and models of random processes, Proc. Sixth Berkeley Symp. on Math. Statist. and Prob., 3 (1972), 263-294. 

[28]

S. A. MohammedT. S. Zhang and H. Z. Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Mem. Am. Math. Soc., 196 (2008), 1-105.  doi: 10.1090/memo/0917.

[29]

S. Nakao, On weak convergence of sequences of continuous local martingale, Ann. Inst. H. Poincaré Probab. Statist., 22 (1986), 371-380. 

[30]

P. Protter, Approximations of solutions of stochastic differential equations driven by semimartingales, Ann. Probab., 13 (1985), 716-743.  doi: 10.1214/aop/1176992905.

[31]

N. V. Krylov and B. L. Rozovski$\check{{\rm i}}$, Stochastic evolution equations, Current Problems in Mathematics, Vol. 14 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 256 (1979), 71–147.

[32]

J. Shen and K. N. Lu, Wong-Zakai approximations and center manifolds of stochastic differential equations, J. Differ. Equ., 263 (2017), 4929-4977.  doi: 10.1016/j.jde.2017.06.005.

[33]

J. ShenK. N. Lu and W. N. Zhang, Heteroclinic chaotic behavior driven by a Brownian motion, J. Differ. Equ., 255 (2013), 4185-4225.  doi: 10.1016/j.jde.2013.08.003.

[34]

D. W. Stook and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, Proc. Sixth Berkeley Symp. on Math. Statist. and Prob., 3 (1972), 333-359. 

[35]

H. J. Sussmann, An interpretation of stochastic differential equations as ordinary differential equations which depend on the sample point, Bull. Amer. Math. Soc., 83 (1977), 296-298.  doi: 10.1090/S0002-9904-1977-14312-7.

[36]

H. J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. Probab., 6 (1978), 19-41.  doi: 10.1214/aop/1176995608.

[37]

G. Tessitore and J. Zabczyk, Wong-Zakai approximation of stochastic evolution equations, J. Evol. Equ., 6 (2006), 621-655.  doi: 10.1007/s00028-006-0280-9.

[38]

X. H. WangK. N. Lu and B. X. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equ., 264 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.

[39]

W. Wang and J. Q. Duan, A dynamical approximation for stochastic partial differential equations, J. Math. Phys., 48 (2007), 102701. doi: 10.1063/1.2800164.

[40]

E. Wong and M. Zakai, On the relation between ordinary and stochastic differential equations, Int. J. Eng. Sci., 3 (1965), 213-229.  doi: 10.1016/0020-7225(65)90045-5.

[41]

E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Stat., 36 (1965), 1560-1564.  doi: 10.1214/aoms/1177699916.

[42]

X. T. YanX. M. Liu and M. H. Yang, Random attractors of stochastic partial differential equations: A smooth approximation approach, Stoch. Anal. Appl., 35 (2017), 1007-1029.  doi: 10.1080/07362994.2017.1345317.

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