doi: 10.3934/dcds.2021176
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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 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 & Continuous Dynamical Systems, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

[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.  Google Scholar

[30]

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

[31]

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

[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.  Google Scholar

[33]

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

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

[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.  Google Scholar

[30]

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

[31]

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

[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.  Google Scholar

[33]

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

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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