Random dynamical system generated by nonautonomous stochastic differential equations driven by fractional Brownian motions
Main Article Content
Abstract
In this paper, we prove that a non-autonomous stochastic differential equation generates a continuous random dynamical system. The flow then possesses a random pullback attractor under the dissipativity condition(s) of the drift and smallness of diffusion part.
Article Details
Keywords
stochastic differential equations (SDE), Young integrals, Bebutov flow,, random dynamical systems, random attractors
References
[1] L. Arnold, Random Dynamical Systems. Springer, Berlin Heidelberg New York, 1998.
[2] I. Bailleul, S. Riedel, M. Scheutzow, Random dynamical systems, rough paths and rough flows. J. Differential Equations, Vol. 262, (2017), 5792–5823.
[3] C. Castaing, M. Valadier , Convex analysis and measurable multifunctions. Lecture Notes in Math. No 580, Springer-Verlag, Berlin, Heidelberg, New York, 1977
[4] N. D. Cong, L. H. Duc, P. T. Hong, Young differential equations revisited. J. Dyn. Diff. Equat., Vol. 30, Iss. 4, (2018), 1921–1943.
[5] H. Crauel, P. Kloeden, Nonautonomous and random attractors. Jahresber Dtsch. Math-Ver. 117 (2015), 173–206.
[6] P. Kloeden, M. Rasmussen, Nonautonomous Dynamical System. Mathematical Surveys and Monographs. https://doi.org/10.1090/surv/176 (2011).
[7] L. H. Duc, P. T. Hong, N. D. Cong, Asymptotic stability for stochastic dissipative systems with a H¨older noise. SIAM J. Control Optim.,Vol. 57, No. 4, pp. 3046 - 3071, (2019)
[8] L. H. Duc, P. T. Hong, Asymptotic Dynamics of Young Differential Equations . J. Dyn. Diff. Equat. 35, 1667–1692 (2023).
[9] L. H. Duc, B. Schmalfuß, S. Siegmund, A note on the generation of random dynamical systems from fractional stochastic delay differential equations. Stoch. Dyn., Vol. 15(2015), No. 3, 1–13.
[10] L. H. Duc, Random attractors for dissipative systems with rough noises Discrete & Continuous Dynamical Systems 2022, 42(4): 1873-1902
[11] P. Friz, N. Victoir, Multidimensional stochastic processes as rough paths: theory and applications. Cambridge Unversity Press, Cambridge, 2010.
[12] M. Garrido-Atienza, B. Maslowski, B. Schmalfuß, Random attractors for stochastic equations driven by a fractional Brownian motion. International Journal of Bifurcation and Chaos, Vol.20, No. 9 (2010) 2761–2782.
[13] M. Garrido-Atienza, B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion. J. Dyn. Diff. Equat., 23, (2011), 671–681. DOI 10.1007/s10884-011-9222-5.
[14] P. T. Hong, Nonautonomous attractors for Young differential equations driven by unbounded variation paths. Journal of Mathematics and Science Thang Long University, Vol. 1, No. 4 (2022).
[15] Y. Hu, Analysis on Gaussian Spaces. World scientific Publishing, (2016).
[16] T. Lyons, Differential equations driven by rough signals, I, An extension of an inequality of L.C. Young. Math. Res. Lett. 1, (1994), 451–464.
[17] B. Mandelbrot, J. van Ness, Fractional Brownian motion, fractional noises and applications. SIAM Review, 4, No. 10, (1968), 422–437.
[18] V. V. Nemytskii, V. V. Stepanov, Qualitative Theory of Differential Equations. GITTL, Moscow–Leningrad. (1949). English translation, Princeton University Press, (1960).
[19] R. J. Sacker, G. Sell, Lifting properties in skew-product flows with applications to differential equations. Memoirs of the American Mathematical Society, Vol. 11, No. 190, (1977).
[20] B. Schmalfuß, Attractors for the nonautonomous dynamical systems. In K. Groger, B. Fiedler and J. Sprekels, editors, Proceedings EQUADIFF99, World Scientific, (2000), 684–690.
[21] B. Schmalfuß. Attractors for the nonautonomous and random dynamical systems perturbed by impulses. Discret. Contin. Dyn. Syst., 9, No. 3, (2003), 727–744.
[22] G. Sell, Nonautonomous differential equations and topological dynamics. I. The Basic theory Transactions of the American Mathematical Society, Vol. 127, No. 2 (1967), 241-262.
[23] G. Sell, Nonautonomous Differential Equations and Topological Dynamics. II. Limiting Equations Transactions of the American Mathematical Society, Vol. 127, No. 2 ( 1967), 263-283
[24] L.C. Young, An integration of H¨older type, connected with Stieltjes integration. Acta Math. 67, (1936), 251–282.
[2] I. Bailleul, S. Riedel, M. Scheutzow, Random dynamical systems, rough paths and rough flows. J. Differential Equations, Vol. 262, (2017), 5792–5823.
[3] C. Castaing, M. Valadier , Convex analysis and measurable multifunctions. Lecture Notes in Math. No 580, Springer-Verlag, Berlin, Heidelberg, New York, 1977
[4] N. D. Cong, L. H. Duc, P. T. Hong, Young differential equations revisited. J. Dyn. Diff. Equat., Vol. 30, Iss. 4, (2018), 1921–1943.
[5] H. Crauel, P. Kloeden, Nonautonomous and random attractors. Jahresber Dtsch. Math-Ver. 117 (2015), 173–206.
[6] P. Kloeden, M. Rasmussen, Nonautonomous Dynamical System. Mathematical Surveys and Monographs. https://doi.org/10.1090/surv/176 (2011).
[7] L. H. Duc, P. T. Hong, N. D. Cong, Asymptotic stability for stochastic dissipative systems with a H¨older noise. SIAM J. Control Optim.,Vol. 57, No. 4, pp. 3046 - 3071, (2019)
[8] L. H. Duc, P. T. Hong, Asymptotic Dynamics of Young Differential Equations . J. Dyn. Diff. Equat. 35, 1667–1692 (2023).
[9] L. H. Duc, B. Schmalfuß, S. Siegmund, A note on the generation of random dynamical systems from fractional stochastic delay differential equations. Stoch. Dyn., Vol. 15(2015), No. 3, 1–13.
[10] L. H. Duc, Random attractors for dissipative systems with rough noises Discrete & Continuous Dynamical Systems 2022, 42(4): 1873-1902
[11] P. Friz, N. Victoir, Multidimensional stochastic processes as rough paths: theory and applications. Cambridge Unversity Press, Cambridge, 2010.
[12] M. Garrido-Atienza, B. Maslowski, B. Schmalfuß, Random attractors for stochastic equations driven by a fractional Brownian motion. International Journal of Bifurcation and Chaos, Vol.20, No. 9 (2010) 2761–2782.
[13] M. Garrido-Atienza, B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion. J. Dyn. Diff. Equat., 23, (2011), 671–681. DOI 10.1007/s10884-011-9222-5.
[14] P. T. Hong, Nonautonomous attractors for Young differential equations driven by unbounded variation paths. Journal of Mathematics and Science Thang Long University, Vol. 1, No. 4 (2022).
[15] Y. Hu, Analysis on Gaussian Spaces. World scientific Publishing, (2016).
[16] T. Lyons, Differential equations driven by rough signals, I, An extension of an inequality of L.C. Young. Math. Res. Lett. 1, (1994), 451–464.
[17] B. Mandelbrot, J. van Ness, Fractional Brownian motion, fractional noises and applications. SIAM Review, 4, No. 10, (1968), 422–437.
[18] V. V. Nemytskii, V. V. Stepanov, Qualitative Theory of Differential Equations. GITTL, Moscow–Leningrad. (1949). English translation, Princeton University Press, (1960).
[19] R. J. Sacker, G. Sell, Lifting properties in skew-product flows with applications to differential equations. Memoirs of the American Mathematical Society, Vol. 11, No. 190, (1977).
[20] B. Schmalfuß, Attractors for the nonautonomous dynamical systems. In K. Groger, B. Fiedler and J. Sprekels, editors, Proceedings EQUADIFF99, World Scientific, (2000), 684–690.
[21] B. Schmalfuß. Attractors for the nonautonomous and random dynamical systems perturbed by impulses. Discret. Contin. Dyn. Syst., 9, No. 3, (2003), 727–744.
[22] G. Sell, Nonautonomous differential equations and topological dynamics. I. The Basic theory Transactions of the American Mathematical Society, Vol. 127, No. 2 (1967), 241-262.
[23] G. Sell, Nonautonomous Differential Equations and Topological Dynamics. II. Limiting Equations Transactions of the American Mathematical Society, Vol. 127, No. 2 ( 1967), 263-283
[24] L.C. Young, An integration of H¨older type, connected with Stieltjes integration. Acta Math. 67, (1936), 251–282.