• Previous Article
    Global strong solution to the two dimensional nonhomogeneous incompressible heat conducting Navier-Stokes flows with vacuum
  • DCDS-B Home
  • This Issue
  • Next Article
    On global large energy solutions to the viscous shallow water equations
November  2020, 25(11): 4295-4316. doi: 10.3934/dcdsb.2020098

On the approaching time towards the attractor of differential equations perturbed by small noise

Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany

Received  August 2019 Revised  November 2019 Published  April 2020

We estimate the time that a point or set, respectively, requires to approach the attractor of a radially symmetric gradient type stochastic differential equation driven by small noise. Here, both of these times tend to infinity as the noise gets small. However, the rates at which they go to infinity differ significantly. In the case of a set approaching the attractor, we use large deviation techniques to show that this time increases exponentially. In the case of a point approaching the attractor, we apply a time change and compare the accelerated process to a process on the sphere and obtain that this time increases merely linearly.

Citation: Isabell Vorkastner. On the approaching time towards the attractor of differential equations perturbed by small noise. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4295-4316. doi: 10.3934/dcdsb.2020098
References:
[1]

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

[2]

P. H. Baxendale, Asymptotic behaviour of stochastic flows of diffeomorphisms, in Stochastic Processes and Their Applications (Nagoya, 1985), Lecture Notes in Math., Vol. 1203, Springer, Berlin, 1986, 1–19. doi: 10.1007/BFb0076869.  Google Scholar

[3]

T. CaraballoI. D. Chueshov and P. E. Kloeden, Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain, SIAM J. Math. Anal., 38 (2006/07), 1489-1507.  doi: 10.1137/050647281.  Google Scholar

[4]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math., Vol. 580, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[5]

H. Crauel and M. Scheutzow, Minimal random attractors, J. Differential Equations, 265 (2018), 702-718.  doi: 10.1016/j.jde.2018.03.011.  Google Scholar

[6]

A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, 2$^{nd}$ edition, Applications of Mathematics, Vol. 38, Springer-Verlag, New York, 1998.  Google Scholar

[7]

G. Dimitroff and M. Scheutzow, Attractors and expansion for Brownian flows, Electron. J. Probab., 16 (2011), 1193-1213.  doi: 10.1214/EJP.v16-894.  Google Scholar

[8]

F. FlandoliB. Gess and M. Scheutzow, Synchronization by noise, Probab. Theory Related Fields, 168 (2017), 511-556.  doi: 10.1007/s00440-016-0716-2.  Google Scholar

[9]

F. FlandoliB. Gess and M. Scheutzow, Synchronization by noise for order-preserving random dynamical systems, Ann. Probab., 45 (2017), 1325-1350.  doi: 10.1214/16-AOP1088.  Google Scholar

[10]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Grundlehren der Mathematischen Wissenschaften, Vol. 260, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4684-0176-9.  Google Scholar

[11]

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

[12]

A. J. Homburg, Synchronization in minimal iterated function systems on compact manifolds, Bull. Braz. Math. Soc. (N.S.), 49 (2018), 615-635.  doi: 10.1007/s00574-018-0073-0.  Google Scholar

[13]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2$^nd$ edition, Graduate Texts in Mathematics, Vol. 113, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[14]

F. Martinelli and E. Scoppola, Small random perturbations of dynamical systems: Exponential loss of memory of the initial condition, Comm. Math. Phys., 120 (1988), 25-69.  doi: 10.1007/BF01223205.  Google Scholar

[15]

O. M. Tearne, Collapse of attractors for ODEs under small random perturbations, Probab. Theory Related Fields, 141 (2008), 1-18.  doi: 10.1007/s00440-006-0051-0.  Google Scholar

[16]

I. Vorkastner, Noise dependent synchronization of a degenerate SDE, Stoch. Dyn., 18 (2018), 1850007, 21pp. doi: 10.1142/S0219493718500077.  Google Scholar

show all references

References:
[1]

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

[2]

P. H. Baxendale, Asymptotic behaviour of stochastic flows of diffeomorphisms, in Stochastic Processes and Their Applications (Nagoya, 1985), Lecture Notes in Math., Vol. 1203, Springer, Berlin, 1986, 1–19. doi: 10.1007/BFb0076869.  Google Scholar

[3]

T. CaraballoI. D. Chueshov and P. E. Kloeden, Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain, SIAM J. Math. Anal., 38 (2006/07), 1489-1507.  doi: 10.1137/050647281.  Google Scholar

[4]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math., Vol. 580, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[5]

H. Crauel and M. Scheutzow, Minimal random attractors, J. Differential Equations, 265 (2018), 702-718.  doi: 10.1016/j.jde.2018.03.011.  Google Scholar

[6]

A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, 2$^{nd}$ edition, Applications of Mathematics, Vol. 38, Springer-Verlag, New York, 1998.  Google Scholar

[7]

G. Dimitroff and M. Scheutzow, Attractors and expansion for Brownian flows, Electron. J. Probab., 16 (2011), 1193-1213.  doi: 10.1214/EJP.v16-894.  Google Scholar

[8]

F. FlandoliB. Gess and M. Scheutzow, Synchronization by noise, Probab. Theory Related Fields, 168 (2017), 511-556.  doi: 10.1007/s00440-016-0716-2.  Google Scholar

[9]

F. FlandoliB. Gess and M. Scheutzow, Synchronization by noise for order-preserving random dynamical systems, Ann. Probab., 45 (2017), 1325-1350.  doi: 10.1214/16-AOP1088.  Google Scholar

[10]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Grundlehren der Mathematischen Wissenschaften, Vol. 260, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4684-0176-9.  Google Scholar

[11]

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

[12]

A. J. Homburg, Synchronization in minimal iterated function systems on compact manifolds, Bull. Braz. Math. Soc. (N.S.), 49 (2018), 615-635.  doi: 10.1007/s00574-018-0073-0.  Google Scholar

[13]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2$^nd$ edition, Graduate Texts in Mathematics, Vol. 113, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[14]

F. Martinelli and E. Scoppola, Small random perturbations of dynamical systems: Exponential loss of memory of the initial condition, Comm. Math. Phys., 120 (1988), 25-69.  doi: 10.1007/BF01223205.  Google Scholar

[15]

O. M. Tearne, Collapse of attractors for ODEs under small random perturbations, Probab. Theory Related Fields, 141 (2008), 1-18.  doi: 10.1007/s00440-006-0051-0.  Google Scholar

[16]

I. Vorkastner, Noise dependent synchronization of a degenerate SDE, Stoch. Dyn., 18 (2018), 1850007, 21pp. doi: 10.1142/S0219493718500077.  Google Scholar

Figure 1.  Outline of the set $ |X_t^\varepsilon(S_{r_2})| $ and the stopping times $ \sigma_n $ and $ \rho_n $
Figure 2.  Outline of the semi-flow $ F(g^\alpha) $ in $ \mathbb{R}^2 $ at time $ t $
[1]

Jicheng Liu, Meiling Zhao. Normal deviation of synchronization of stochastic coupled systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021079

[2]

Scott Schmieding, Rodrigo Treviño. Random substitution tilings and deviation phenomena. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3869-3902. doi: 10.3934/dcds.2021020

[3]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

[4]

Xuping Zhang. Pullback random attractors for fractional stochastic $ p $-Laplacian equation with delay and multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021107

[5]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175

[6]

Seddigheh Banihashemi, Hossein Jafaria, Afshin Babaei. A novel collocation approach to solve a nonlinear stochastic differential equation of fractional order involving a constant delay. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021025

[7]

Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021031

[8]

Jiacheng Wang, Peng-Fei Yao. On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021043

[9]

Quan Hai, Shutang Liu. Mean-square delay-distribution-dependent exponential synchronization of chaotic neural networks with mixed random time-varying delays and restricted disturbances. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3097-3118. doi: 10.3934/dcdsb.2020221

[10]

Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1059-1076. doi: 10.3934/cpaa.2021006

[11]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[12]

Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192

[13]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209

[14]

Mrinal K. Ghosh, Somnath Pradhan. A nonzero-sum risk-sensitive stochastic differential game in the orthant. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021025

[15]

Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021020

[16]

Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113

[17]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[18]

Lingyu Li, Zhang Chen. Asymptotic behavior of non-autonomous random Ginzburg-Landau equation driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3303-3333. doi: 10.3934/dcdsb.2020233

[19]

Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021026

[20]

Wensheng Yin, Jinde Cao, Guoqiang Zheng. Further results on stabilization of stochastic differential equations with delayed feedback control under $ G $-expectation framework. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021072

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (84)
  • HTML views (238)
  • Cited by (0)

Other articles
by authors

[Back to Top]