• Previous Article
    Instability of the standing waves for a Benney-Roskes/Zakharov-Rubenchik system and blow-up for the zakharov equations
  • DCDS-B Home
  • This Issue
  • Next Article
    Attractors for first order lattice systems with almost periodic nonlinear part
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  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 & Continuous Dynamical Systems - B, 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[29]

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

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[29]

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

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Anhui Gu. Asymptotic behavior of random lattice dynamical systems and their Wong-Zakai approximations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5737-5767. doi: 10.3934/dcdsb.2019104

[2]

Anhui Gu, Kening Lu, Bixiang Wang. Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 185-218. doi: 10.3934/dcds.2019008

[3]

Yanfeng Guo, Jinqiao Duan, Donglong Li. Approximation of random invariant manifolds for a stochastic Swift-Hohenberg equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1701-1715. doi: 10.3934/dcdss.2016071

[4]

Rongmei Cao, Jiangong You. The existence of integrable invariant manifolds of Hamiltonian partial differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 227-234. doi: 10.3934/dcds.2006.16.227

[5]

Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085

[6]

Zhongkai Guo. Invariant foliations for stochastic partial differential equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5203-5219. doi: 10.3934/dcds.2015.35.5203

[7]

Ludwig Arnold, Igor Chueshov. Cooperative random and stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 1-33. doi: 10.3934/dcds.2001.7.1

[8]

Nguyen Thieu Huy, Pham Van Bang. Invariant stable manifolds for partial neutral functional differential equations in admissible spaces on a half-line. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2993-3011. doi: 10.3934/dcdsb.2015.20.2993

[9]

Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Invariant manifolds as pullback attractors of nonautonomous differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 579-596. doi: 10.3934/dcds.2006.15.579

[10]

Ji Li, Kening Lu, Peter W. Bates. Invariant foliations for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3639-3666. doi: 10.3934/dcds.2014.34.3639

[11]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473

[12]

Maciej J. Capiński, Piotr Zgliczyński. Cone conditions and covering relations for topologically normally hyperbolic invariant manifolds. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 641-670. doi: 10.3934/dcds.2011.30.641

[13]

Ivan Werner. Equilibrium states and invariant measures for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1285-1326. doi: 10.3934/dcds.2015.35.1285

[14]

Fuke Wu, Peter E. Kloeden. Mean-square random attractors of stochastic delay differential equations with random delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1715-1734. doi: 10.3934/dcdsb.2013.18.1715

[15]

Sergio Albeverio, Sonia Mazzucchi. Infinite dimensional integrals and partial differential equations for stochastic and quantum phenomena. Journal of Geometric Mechanics, 2019, 11 (2) : 123-137. doi: 10.3934/jgm.2019006

[16]

Viorel Barbu. Existence for nonlinear finite dimensional stochastic differential equations of subgradient type. Mathematical Control & Related Fields, 2018, 8 (3&4) : 501-508. doi: 10.3934/mcrf.2018020

[17]

I. Baldomá, Àlex Haro. One dimensional invariant manifolds of Gevrey type in real-analytic maps. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 295-322. doi: 10.3934/dcdsb.2008.10.295

[18]

Arne Ogrowsky, Björn Schmalfuss. Unstable invariant manifolds for a nonautonomous differential equation with nonautonomous unbounded delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1663-1681. doi: 10.3934/dcdsb.2013.18.1663

[19]

Rovella Alvaro, Vilamajó Francesc, Romero Neptalí. Invariant manifolds for delay endomorphisms. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 35-50. doi: 10.3934/dcds.2001.7.35

[20]

Graeme D. Chalmers, Desmond J. Higham. Convergence and stability analysis for implicit simulations of stochastic differential equations with random jump magnitudes. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 47-64. doi: 10.3934/dcdsb.2008.9.47

2018 Impact Factor: 1.008

Article outline

[Back to Top]