April  2022, 42(4): 1873-1902. doi: 10.3934/dcds.2021176

Random attractors for dissipative systems with rough noises

1. 

Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany

2. 

Institute of Mathematics, Vietnam Academy of Science and Technology, 10307 Hanoi, Vietnam

 

Received  December 2020 Revised  September 2021 Published  April 2022 Early access  November 2021

We provide an analytic approach to study the asymptotic dynamics of rough differential equations, with the driving noises of Hölder continuity. Such systems can be solved with Lyons' theory of rough paths, in particular the rough integrals are understood in the Gubinelli sense for controlled rough paths. Using the framework of random dynamical systems and random attractors, we prove the existence and upper semi-continuity of the global pullback attractor for dissipative systems perturbed by bounded noises. Moreover, if the unperturbed system is strictly dissipative then the random attractor is a singleton for sufficiently small noise intensity.

Citation: Luu Hoang Duc. Random attractors for dissipative systems with rough noises. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1873-1902. doi: 10.3934/dcds.2021176
References:
[1]

L. Arnold, Random Dynamical Systems, Springer, Berlin Heidelberg New York, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

I. BailleulS. Riedel and M. Scheutzow, Random dynamical systems, rough paths and rough flows, J. Differential Equations, 262 (2017), 5792-5823.  doi: 10.1016/j.jde.2017.02.014.

[3]

T. CaraballoJ. A. Langa and J. C. Robinson, Upper semicontinuity of attractors for small random perturbations of dynamical systems, Commu. Partial Differential Equations, 23 (1998), 1557-1581.  doi: 10.1080/03605309808821394.

[4]

A. N. Carvalho and J. A. Langa, An extension of the concept of gradient semigroups which is stable under perturbation, J. Differential Equations, 246 (2009), 2646-2668.  doi: 10.1016/j.jde.2009.01.007.

[5]

T. CassC. Litterer and T. Lyons, Integrability and tail estimates for Gaussian rough differential equations, Annals of Probability, 41 (2013), 3026-3050.  doi: 10.1214/12-AOP821.

[6]

N. D. CongL. H. Duc and P. T. Hong, Nonautonomous young differential equations revisited, J. Dyn. Diff. Equat., 30 (2018), 1921-1943.  doi: 10.1007/s10884-017-9634-y.

[7]

N. D. Cong, L. H. Duc and P. T. Hong, Pullback attractors for stochastic Young differential delay equations, J. Dyn. Diff. Equat., (2020). doi: 10.1007/s10884-020-09894-9.

[8]

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

[9]

H. Crauel and P. E. Kloeden, Nonautonomous and random attractors, Jahresber Dtsch. Math-Ver., 117 (2015), 173-206.  doi: 10.1365/s13291-015-0115-0.

[10]

L. H. Duc, Controlled differential equations as rough integrals, to appear in Pure and Applied Functional Analysis, Preprint arXiv: 2007.06295.

[11]

L. H. DucM. J. Garrido-AtienzaA. Neuenkirch and B. Schmalfuß, Exponential stability of stochastic evolution equations driven by small fractional Brownian motion with Hurst parameter in $(\frac{1}{2}, 1)$, J. Differential Equations, 264 (2018), 1119-1145.  doi: 10.1016/j.jde.2017.09.033.

[12]

L. H. DucP. T. Hong and N. D. Cong, Asymptotic stability for stochastic dissipative systems with a Hölder noise, SIAM Journal on Control and Optimization, 57 (2019), 3046-3071.  doi: 10.1137/19M1236527.

[13]

P. K. Friz and M. Hairer, A Course on Rough Path with an Introduction to Regularity Structure, Universitext, Vol. XIV, Springer, Cham, 2014. doi: 10.1007/978-3-319-08332-2.

[14]

P. K. Friz and N. B. Victoir, Multidimensional Stochastic Processes as Rough Paths: Theory and Applications, Cambridge Studies in Advanced Mathematics, 120. Cambridge Unversity Press, Cambridge, 2010. doi: 10.1017/CBO9780511845079.

[15]

M. J. Garrido-AtienzaK. Lu and B. Schmalfuß, Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H \in (\frac{1}{3}, \frac{1}{2}]$, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2553-2581.  doi: 10.3934/dcdsb.2015.20.2553.

[16]

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

[17]

M. Gubinelli, Controlling rough paths, J. Funtional Analysis, 216 (2004), 86-140.  doi: 10.1016/j.jfa.2004.01.002.

[18]

M. Hairer, Ergodicity of stochastic differential equations driven by fractional Brownian motion, The Annals of Probability, 33 (2005), 703-758.  doi: 10.1214/009117904000000892.

[19]

M. Hairer and A. Ohashi, Ergodic theory for sdes with extrinsic memory, The Annals of Probability, 35 (2007), 1950-1977.  doi: 10.1214/009117906000001141.

[20]

M. Hairer and N. S. Pillai, Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion, Ann. Inst. Henri Poincaré Probab. Stat., 47 (2011), 601-628.  doi: 10.1214/10-AIHP377.

[21]

M. Hairer and N. S. Pillai, Regularity of laws and ergodicity of hypoelliptic stochastic differential equations driven by rough paths, The Annals of Probability, 41 (2013), 2544-2598.  doi: 10.1214/12-AOP777.

[22]

J. K. Hale and H. Koçak, Dynamics and Bifurcations, Springer-Verlag, 1991. doi: 10.1007/978-1-4612-4426-4.

[23]

J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbedhyperbolic equation, J. Differential Equations, 73 (1988), 197-214.  doi: 10.1016/0022-0396(88)90104-0.

[24]

R. Hesse and A. Neamtu, Local mild solutions for rough stochastic partial differential equations, J. Differential Equations, 267 (2019), 6480-6538.  doi: 10.1016/j.jde.2019.06.026.

[25]

Y. Hu and D. Nualart, Rough path analysis via fractional calculus, Trans. Amer. Math. Soc., 361 (2009), 2689-2718.  doi: 10.1090/S0002-9947-08-04631-X.

[26]

P. Imkeller and B. Schmalfuss, The conjugacy of stochastic and random differential equations and the existence of global attractors, J. Dyn. Diff. Equat., 13 (2001), 215-249.  doi: 10.1023/A:1016673307045.

[27]

H. Keller and B. Schmalfuss, Attractors for stochastic differential equations with nontrivial noise, Bul. Acad. Ştiinţe Repub. Mold. Mat., 26 (1998), 43-54. 

[28]

R. Khasminskii, Stochastic Stability of Differential Equations, 2nd edition, Vol. 66, 2012. doi: 10.1007/978-3-642-23280-0.

[29]

M. Ledoux, Isoperimetry and Gaussian analysis, In Lectures on Probability Theory and Statistics (Saint-Flour)., Lecture Notes in Math., 1648, Springer, Berlin, 1996,165–294. doi: 10.1007/BFb0095676.

[30]

T. J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoam., 14 (1998), 215-310.  doi: 10.4171/RMI/240.

[31]

T. J. Lyons, M. Caruana and T. Lévy, Differential Equations Driven by Rough Paths, Lecture Notes in Mathematics, Vol. 1908, Springer, Berlin, 2007.

[32]

B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motion, fractional noises and applications, SIAM Review, 10 (1968), 422-437.  doi: 10.1137/1010093.

[33]

X. Mao, Exponential Stability of Stochastic Differential Equations, NewYork: Marcel Dekker, 1994.

[34]

I. Nourdin, Selected Aspects of Fractional Brownian Motion, Bocconi University Press, Springer, 2012. doi: 10.1007/978-88-470-2823-4.

[35]

S. Riedel and M. Scheutzow, Rough differential equations with unbounded drift terms, J. Differential Equations, 262 (2017), 283-312.  doi: 10.1016/j.jde.2016.09.021.

[36]

A. V. Skorohod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Translations of Mathematical Monographs, American Mathematical Society, Vol. 78, 1989. doi: 10.1090/mmono/078.

[37]

H. J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, The Annals of Probability, 6 (1978), 19-41.  doi: 10.1214/aop/1176995608.

[38]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31 pp. doi: 10.1142/S0219493714500099.

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer, Berlin Heidelberg New York, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

I. BailleulS. Riedel and M. Scheutzow, Random dynamical systems, rough paths and rough flows, J. Differential Equations, 262 (2017), 5792-5823.  doi: 10.1016/j.jde.2017.02.014.

[3]

T. CaraballoJ. A. Langa and J. C. Robinson, Upper semicontinuity of attractors for small random perturbations of dynamical systems, Commu. Partial Differential Equations, 23 (1998), 1557-1581.  doi: 10.1080/03605309808821394.

[4]

A. N. Carvalho and J. A. Langa, An extension of the concept of gradient semigroups which is stable under perturbation, J. Differential Equations, 246 (2009), 2646-2668.  doi: 10.1016/j.jde.2009.01.007.

[5]

T. CassC. Litterer and T. Lyons, Integrability and tail estimates for Gaussian rough differential equations, Annals of Probability, 41 (2013), 3026-3050.  doi: 10.1214/12-AOP821.

[6]

N. D. CongL. H. Duc and P. T. Hong, Nonautonomous young differential equations revisited, J. Dyn. Diff. Equat., 30 (2018), 1921-1943.  doi: 10.1007/s10884-017-9634-y.

[7]

N. D. Cong, L. H. Duc and P. T. Hong, Pullback attractors for stochastic Young differential delay equations, J. Dyn. Diff. Equat., (2020). doi: 10.1007/s10884-020-09894-9.

[8]

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

[9]

H. Crauel and P. E. Kloeden, Nonautonomous and random attractors, Jahresber Dtsch. Math-Ver., 117 (2015), 173-206.  doi: 10.1365/s13291-015-0115-0.

[10]

L. H. Duc, Controlled differential equations as rough integrals, to appear in Pure and Applied Functional Analysis, Preprint arXiv: 2007.06295.

[11]

L. H. DucM. J. Garrido-AtienzaA. Neuenkirch and B. Schmalfuß, Exponential stability of stochastic evolution equations driven by small fractional Brownian motion with Hurst parameter in $(\frac{1}{2}, 1)$, J. Differential Equations, 264 (2018), 1119-1145.  doi: 10.1016/j.jde.2017.09.033.

[12]

L. H. DucP. T. Hong and N. D. Cong, Asymptotic stability for stochastic dissipative systems with a Hölder noise, SIAM Journal on Control and Optimization, 57 (2019), 3046-3071.  doi: 10.1137/19M1236527.

[13]

P. K. Friz and M. Hairer, A Course on Rough Path with an Introduction to Regularity Structure, Universitext, Vol. XIV, Springer, Cham, 2014. doi: 10.1007/978-3-319-08332-2.

[14]

P. K. Friz and N. B. Victoir, Multidimensional Stochastic Processes as Rough Paths: Theory and Applications, Cambridge Studies in Advanced Mathematics, 120. Cambridge Unversity Press, Cambridge, 2010. doi: 10.1017/CBO9780511845079.

[15]

M. J. Garrido-AtienzaK. Lu and B. Schmalfuß, Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H \in (\frac{1}{3}, \frac{1}{2}]$, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2553-2581.  doi: 10.3934/dcdsb.2015.20.2553.

[16]

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

[17]

M. Gubinelli, Controlling rough paths, J. Funtional Analysis, 216 (2004), 86-140.  doi: 10.1016/j.jfa.2004.01.002.

[18]

M. Hairer, Ergodicity of stochastic differential equations driven by fractional Brownian motion, The Annals of Probability, 33 (2005), 703-758.  doi: 10.1214/009117904000000892.

[19]

M. Hairer and A. Ohashi, Ergodic theory for sdes with extrinsic memory, The Annals of Probability, 35 (2007), 1950-1977.  doi: 10.1214/009117906000001141.

[20]

M. Hairer and N. S. Pillai, Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion, Ann. Inst. Henri Poincaré Probab. Stat., 47 (2011), 601-628.  doi: 10.1214/10-AIHP377.

[21]

M. Hairer and N. S. Pillai, Regularity of laws and ergodicity of hypoelliptic stochastic differential equations driven by rough paths, The Annals of Probability, 41 (2013), 2544-2598.  doi: 10.1214/12-AOP777.

[22]

J. K. Hale and H. Koçak, Dynamics and Bifurcations, Springer-Verlag, 1991. doi: 10.1007/978-1-4612-4426-4.

[23]

J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbedhyperbolic equation, J. Differential Equations, 73 (1988), 197-214.  doi: 10.1016/0022-0396(88)90104-0.

[24]

R. Hesse and A. Neamtu, Local mild solutions for rough stochastic partial differential equations, J. Differential Equations, 267 (2019), 6480-6538.  doi: 10.1016/j.jde.2019.06.026.

[25]

Y. Hu and D. Nualart, Rough path analysis via fractional calculus, Trans. Amer. Math. Soc., 361 (2009), 2689-2718.  doi: 10.1090/S0002-9947-08-04631-X.

[26]

P. Imkeller and B. Schmalfuss, The conjugacy of stochastic and random differential equations and the existence of global attractors, J. Dyn. Diff. Equat., 13 (2001), 215-249.  doi: 10.1023/A:1016673307045.

[27]

H. Keller and B. Schmalfuss, Attractors for stochastic differential equations with nontrivial noise, Bul. Acad. Ştiinţe Repub. Mold. Mat., 26 (1998), 43-54. 

[28]

R. Khasminskii, Stochastic Stability of Differential Equations, 2nd edition, Vol. 66, 2012. doi: 10.1007/978-3-642-23280-0.

[29]

M. Ledoux, Isoperimetry and Gaussian analysis, In Lectures on Probability Theory and Statistics (Saint-Flour)., Lecture Notes in Math., 1648, Springer, Berlin, 1996,165–294. doi: 10.1007/BFb0095676.

[30]

T. J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoam., 14 (1998), 215-310.  doi: 10.4171/RMI/240.

[31]

T. J. Lyons, M. Caruana and T. Lévy, Differential Equations Driven by Rough Paths, Lecture Notes in Mathematics, Vol. 1908, Springer, Berlin, 2007.

[32]

B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motion, fractional noises and applications, SIAM Review, 10 (1968), 422-437.  doi: 10.1137/1010093.

[33]

X. Mao, Exponential Stability of Stochastic Differential Equations, NewYork: Marcel Dekker, 1994.

[34]

I. Nourdin, Selected Aspects of Fractional Brownian Motion, Bocconi University Press, Springer, 2012. doi: 10.1007/978-88-470-2823-4.

[35]

S. Riedel and M. Scheutzow, Rough differential equations with unbounded drift terms, J. Differential Equations, 262 (2017), 283-312.  doi: 10.1016/j.jde.2016.09.021.

[36]

A. V. Skorohod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Translations of Mathematical Monographs, American Mathematical Society, Vol. 78, 1989. doi: 10.1090/mmono/078.

[37]

H. J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, The Annals of Probability, 6 (1978), 19-41.  doi: 10.1214/aop/1176995608.

[38]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31 pp. doi: 10.1142/S0219493714500099.

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