• 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]

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

[2]

Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352

[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]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[5]

Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240

[6]

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

[7]

Stefan Ruschel, Serhiy Yanchuk. The Spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321

[8]

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

[9]

Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270

[10]

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

[11]

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

[12]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[13]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

[14]

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

[15]

Patrick W. Dondl, Martin Jesenko. Threshold phenomenon for homogenized fronts in random elastic media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 353-372. doi: 10.3934/dcdss.2020329

[16]

Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299

[17]

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

[18]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110

[19]

Andreu Ferré Moragues. Properties of multicorrelation sequences and large returns under some ergodicity assumptions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020386

[20]

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

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (61)
  • HTML views (209)
  • Cited by (0)

Other articles
by authors

[Back to Top]