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June  2018, 23(4): 1689-1720. doi: 10.3934/dcdsb.2018072

Asymptotic behavior of random fitzhugh-nagumo systems driven by colored noise

1. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

2. 

Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

Received  July 2017 Revised  August 2017 Published  January 2018

Fund Project: The first author was supported by China Scholarship Council (No. 201608505082)

In this paper, we prove the existence and uniqueness of random attractors for the FitzHugh-Nagumo system driven by colored noise with a nonlinear diffusion term. We demonstrate that the colored noise is much easier to deal with than the white noise for studying the pathwise dynamics of stochastic systems. In addition, we show the attractors of the random FitzHugh-Nagumo system driven by a linear multiplicative colored noise converge to that of the corresponding stochastic system driven by a linear multiplicative white noise.

Citation: Anhui Gu, Bixiang Wang. Asymptotic behavior of random fitzhugh-nagumo systems driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1689-1720. doi: 10.3934/dcdsb.2018072
References:
[1]

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

[2]

A. Adili and B. Wang, Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 643-666. Google Scholar

[3]

V. Anishchenko, V. Astakhov, A. Neiman, T. Vadivasova and L. Schimansky-Geier, Nonlinear Dynamics of Chaotic and Stochastic Systems: Tutorial and Modern Developments, Springer-Verlag, Berlin 2007. Google Scholar

[4]

L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley-Interscience, New York, 1974. Google Scholar

[5]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. Google Scholar

[6]

P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21. doi: 10.1142/S0219493706001621. Google Scholar

[7]

P. W. BatesK. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869. doi: 10.1016/j.jde.2008.05.017. Google Scholar

[8]

W. J. BeynB. GessP. Lescot and M. Röckner, The global random attractor for a class of stochastic porous media equations, Comm. Partial Differential Equations, 36 (2011), 446-469. Google Scholar

[9]

T. CaraballoM. Garrido-AtienzaB. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443. doi: 10.3934/dcds.2008.21.415. Google Scholar

[10]

T. CaraballoJ. Real and I. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 525-539. doi: 10.3934/dcdsb.2008.9.525. Google Scholar

[11]

T. Caraballo and J. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 10 (2003), 491-513. Google Scholar

[12]

T. CaraballoM. Garrido-AtienzaB. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455. doi: 10.3934/dcdsb.2010.14.439. Google Scholar

[13]

T. CaraballoM. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684. doi: 10.1016/j.na.2011.02.047. Google Scholar

[14]

T. CaraballoJ. LangaV. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Analysis, 11 (2003), 153-201. doi: 10.1023/A:1022902802385. Google Scholar

[15]

I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dynamical Systems, 19 (2004), 127-144. doi: 10.1080/1468936042000207792. Google Scholar

[16]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341. doi: 10.1007/BF02219225. Google Scholar

[17]

J. Doob, The Brownian movement and stochastic equations, Annals of Math., 43 (1942), 351-369. doi: 10.2307/1968873. Google Scholar

[18]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. Google Scholar

[19]

J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Comm. Math. Sci., 1 (2003), 133-151. doi: 10.4310/CMS.2003.v1.n1.a9. Google Scholar

[20]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466. doi: 10.1016/S0006-3495(61)86902-6. Google Scholar

[21]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083. Google Scholar

[22]

M. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion, J. Dynam. Differential Equations, 23 (2011), 671-681. doi: 10.1007/s10884-011-9222-5. Google Scholar

[23]

M. Garrido-AtienzaA. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388. doi: 10.1142/S0219493711003358. Google Scholar

[24]

M. Garrido-AtienzaB. Maslowski and B. Schmalfuss, Random attractors for stochastic equations driven by a fractional Brownian motion, International J. Bifurcation and Chaos, 20 (2010), 2761-2782. doi: 10.1142/S0218127410027349. Google Scholar

[25] W. GerstnerW. KistlerR. Naud and L. Paninski, Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition, Cambridge University Press, Cambridge, 2014.
[26]

B. GessW. Liu and M. Rockner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differential Equations, 251 (2011), 1225-1253. doi: 10.1016/j.jde.2011.02.013. Google Scholar

[27]

B. Gess, Random attractors for degenerate stochastic partial differential equations, J. Dyn. Diff. Eqns., 25 (2013), 121-157. doi: 10.1007/s10884-013-9294-5. Google Scholar

[28]

B. Gess, Random attractors for singular stochastic evolution equations, J. Differential Equations, 255 (2013), 524-559. doi: 10.1016/j.jde.2013.04.023. Google Scholar

[29]

P. Hänggi, Colored Noise in Dynamical Systems: A Functional Calculus Approach, in: Noise in Nonlinear Dynamical Systems, vol. 1, F. Moss and P. V. E. McClintock, eds., chap. 9, pp. 307328, Cambridge University Press, 1989.Google Scholar

[30]

P. Häunggi, P. Jung, Colored Noise in Dynamical Systems, in Advances in Chemical Physics, Volume 89 (eds I. Prigogine and S. A. Rice), John Wiley & Sons, Inc., Hoboken, NJ, 1994.Google Scholar

[31]

W. Horsthemke and R. Lefever, Noise-Induced Transitions: Theory and Applications in Physics, Chemistry, and Biology, Springer-Verlag Berlin, 1984. Google Scholar

[32]

J. Huang and W. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains, Discrete Contin. Dyn. Syst., 24 (2009), 855-882. doi: 10.3934/dcds.2009.24.855. Google Scholar

[33]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Noth-Holland, 2nd ed, 1981. Google Scholar

[34]

T. JiangX. Liu and J. Duan, Approximation for random stable manifolds under multiplicative correlated noises, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3163-3174. doi: 10.3934/dcdsb.2016091. Google Scholar

[35]

N. van Kampen, Stochastic Processes in Physics and Chemistry, Amsterdam-New York, 1981. Google Scholar

[36]

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

[37]

P. E. Kloeden and J. Langa, Flattening, squeezing and the existence of random attractors, Proc. Royal Soc. London Serie A., 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753. Google Scholar

[38]

M. Klosek-DygasB. Matkowsky and Z. Schuss, Colored noise in dynamical systems, SIAM J. Appl. Math., 48 (1988), 425-441. doi: 10.1137/0148023. Google Scholar

[39]

K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, submitted.Google Scholar

[40]

Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23. doi: 10.1016/j.jde.2007.10.009. Google Scholar

[41]

J. NagumoS. Arimoto and S. Yosimzawa, An active pulse transmission line simulating nerve axon, Proc. J. R. E., 50 (1962), 2061-2070. doi: 10.1109/JRPROC.1962.288235. Google Scholar

[42]

G. Uhlenbeck and L. Ornstein, On the theory of Brownian motion, Phys. Rev., 36 (1930), 823-841. doi: 10.1103/PhysRev.36.823. Google Scholar

[43] L. RidolfiP. D'Odorico and F. Laio, Noise-Induced Phenomena in the Environmental Sciences, Cambridge University Press, Cambridge, 2011.
[44]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, 185-192, Dresden, 1992.Google Scholar

[45]

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

[46]

D. W. Stook and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, Proc. 6-th Berkeley Symp. on Math. Stat. and Prob., 3 (1972), 333-359. Google Scholar

[47]

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

[48]

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

[49]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. Google Scholar

[50]

B. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537. doi: 10.1016/j.jde.2008.10.012. Google Scholar

[51]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663. doi: 10.1090/S0002-9947-2011-05247-5. Google Scholar

[52]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015. Google Scholar

[53]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn. , 14 (2014), 1450009, 31pp. Google Scholar

[54]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 269-300. Google Scholar

[55]

M. Wang and G. Uhlenbeck, On the theory of Brownian motion. Ⅱ, Rev. Modern Phys., 17 (1945), 323-342. doi: 10.1103/RevModPhys.17.323. Google Scholar

[56]

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

[57]

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

show all references

References:
[1]

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

[2]

A. Adili and B. Wang, Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 643-666. Google Scholar

[3]

V. Anishchenko, V. Astakhov, A. Neiman, T. Vadivasova and L. Schimansky-Geier, Nonlinear Dynamics of Chaotic and Stochastic Systems: Tutorial and Modern Developments, Springer-Verlag, Berlin 2007. Google Scholar

[4]

L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley-Interscience, New York, 1974. Google Scholar

[5]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. Google Scholar

[6]

P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21. doi: 10.1142/S0219493706001621. Google Scholar

[7]

P. W. BatesK. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869. doi: 10.1016/j.jde.2008.05.017. Google Scholar

[8]

W. J. BeynB. GessP. Lescot and M. Röckner, The global random attractor for a class of stochastic porous media equations, Comm. Partial Differential Equations, 36 (2011), 446-469. Google Scholar

[9]

T. CaraballoM. Garrido-AtienzaB. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443. doi: 10.3934/dcds.2008.21.415. Google Scholar

[10]

T. CaraballoJ. Real and I. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 525-539. doi: 10.3934/dcdsb.2008.9.525. Google Scholar

[11]

T. Caraballo and J. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 10 (2003), 491-513. Google Scholar

[12]

T. CaraballoM. Garrido-AtienzaB. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455. doi: 10.3934/dcdsb.2010.14.439. Google Scholar

[13]

T. CaraballoM. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684. doi: 10.1016/j.na.2011.02.047. Google Scholar

[14]

T. CaraballoJ. LangaV. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Analysis, 11 (2003), 153-201. doi: 10.1023/A:1022902802385. Google Scholar

[15]

I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dynamical Systems, 19 (2004), 127-144. doi: 10.1080/1468936042000207792. Google Scholar

[16]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341. doi: 10.1007/BF02219225. Google Scholar

[17]

J. Doob, The Brownian movement and stochastic equations, Annals of Math., 43 (1942), 351-369. doi: 10.2307/1968873. Google Scholar

[18]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. Google Scholar

[19]

J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Comm. Math. Sci., 1 (2003), 133-151. doi: 10.4310/CMS.2003.v1.n1.a9. Google Scholar

[20]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466. doi: 10.1016/S0006-3495(61)86902-6. Google Scholar

[21]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083. Google Scholar

[22]

M. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion, J. Dynam. Differential Equations, 23 (2011), 671-681. doi: 10.1007/s10884-011-9222-5. Google Scholar

[23]

M. Garrido-AtienzaA. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388. doi: 10.1142/S0219493711003358. Google Scholar

[24]

M. Garrido-AtienzaB. Maslowski and B. Schmalfuss, Random attractors for stochastic equations driven by a fractional Brownian motion, International J. Bifurcation and Chaos, 20 (2010), 2761-2782. doi: 10.1142/S0218127410027349. Google Scholar

[25] W. GerstnerW. KistlerR. Naud and L. Paninski, Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition, Cambridge University Press, Cambridge, 2014.
[26]

B. GessW. Liu and M. Rockner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differential Equations, 251 (2011), 1225-1253. doi: 10.1016/j.jde.2011.02.013. Google Scholar

[27]

B. Gess, Random attractors for degenerate stochastic partial differential equations, J. Dyn. Diff. Eqns., 25 (2013), 121-157. doi: 10.1007/s10884-013-9294-5. Google Scholar

[28]

B. Gess, Random attractors for singular stochastic evolution equations, J. Differential Equations, 255 (2013), 524-559. doi: 10.1016/j.jde.2013.04.023. Google Scholar

[29]

P. Hänggi, Colored Noise in Dynamical Systems: A Functional Calculus Approach, in: Noise in Nonlinear Dynamical Systems, vol. 1, F. Moss and P. V. E. McClintock, eds., chap. 9, pp. 307328, Cambridge University Press, 1989.Google Scholar

[30]

P. Häunggi, P. Jung, Colored Noise in Dynamical Systems, in Advances in Chemical Physics, Volume 89 (eds I. Prigogine and S. A. Rice), John Wiley & Sons, Inc., Hoboken, NJ, 1994.Google Scholar

[31]

W. Horsthemke and R. Lefever, Noise-Induced Transitions: Theory and Applications in Physics, Chemistry, and Biology, Springer-Verlag Berlin, 1984. Google Scholar

[32]

J. Huang and W. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains, Discrete Contin. Dyn. Syst., 24 (2009), 855-882. doi: 10.3934/dcds.2009.24.855. Google Scholar

[33]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Noth-Holland, 2nd ed, 1981. Google Scholar

[34]

T. JiangX. Liu and J. Duan, Approximation for random stable manifolds under multiplicative correlated noises, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3163-3174. doi: 10.3934/dcdsb.2016091. Google Scholar

[35]

N. van Kampen, Stochastic Processes in Physics and Chemistry, Amsterdam-New York, 1981. Google Scholar

[36]

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

[37]

P. E. Kloeden and J. Langa, Flattening, squeezing and the existence of random attractors, Proc. Royal Soc. London Serie A., 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753. Google Scholar

[38]

M. Klosek-DygasB. Matkowsky and Z. Schuss, Colored noise in dynamical systems, SIAM J. Appl. Math., 48 (1988), 425-441. doi: 10.1137/0148023. Google Scholar

[39]

K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, submitted.Google Scholar

[40]

Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23. doi: 10.1016/j.jde.2007.10.009. Google Scholar

[41]

J. NagumoS. Arimoto and S. Yosimzawa, An active pulse transmission line simulating nerve axon, Proc. J. R. E., 50 (1962), 2061-2070. doi: 10.1109/JRPROC.1962.288235. Google Scholar

[42]

G. Uhlenbeck and L. Ornstein, On the theory of Brownian motion, Phys. Rev., 36 (1930), 823-841. doi: 10.1103/PhysRev.36.823. Google Scholar

[43] L. RidolfiP. D'Odorico and F. Laio, Noise-Induced Phenomena in the Environmental Sciences, Cambridge University Press, Cambridge, 2011.
[44]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, 185-192, Dresden, 1992.Google Scholar

[45]

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

[46]

D. W. Stook and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, Proc. 6-th Berkeley Symp. on Math. Stat. and Prob., 3 (1972), 333-359. Google Scholar

[47]

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

[48]

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

[49]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. Google Scholar

[50]

B. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537. doi: 10.1016/j.jde.2008.10.012. Google Scholar

[51]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663. doi: 10.1090/S0002-9947-2011-05247-5. Google Scholar

[52]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015. Google Scholar

[53]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn. , 14 (2014), 1450009, 31pp. Google Scholar

[54]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 269-300. Google Scholar

[55]

M. Wang and G. Uhlenbeck, On the theory of Brownian motion. Ⅱ, Rev. Modern Phys., 17 (1945), 323-342. doi: 10.1103/RevModPhys.17.323. Google Scholar

[56]

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

[57]

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

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