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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  November 2020 Early access  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 and 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.

[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.

[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.

[4]

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

[5]

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

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

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.

[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.

[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.

[4]

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

[5]

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

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

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 $
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