Pullback Attractors for Stochastic Young Differential Delay Equations
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Pullback Attractors for Stochastic Young Differential Delay Equations Nguyen Dinh Cong1 · Luu Hoang Duc1,2 · Phan Thanh Hong3
Received: 19 May 2020 / Revised: 30 July 2020 / Accepted: 27 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We study the asymptotic dynamics of stochastic Young differential delay equations under the regular assumptions on Lipschitz continuity of the coefficient functions. Our main results show that, if there is a linear part in the drift term which has no delay factor and has eigenvalues of negative real parts, then the generated random dynamical system possesses a random pullback attractor provided that the Lipschitz coefficients of the remaining parts are small. Keywords Stochastic differential equations (SDE) · Young integral · Random dynamical systems · Random attractors · Exponential stability
1 Introduction Consider the stochastic differential delay equation of the form dy(t) = [Ay(t) + f (yt )]dt + g(yt )d Z (t), y0 = η ∈ C 0,β0 ([−r , 0], Rd ) ⊂ Cr := C ([−r , 0], Rd ), R+ ,
(1.1)
Rd ,
yt is defined by yt : [−r , 0] → yt (s) = y(t + s) for s ∈ [−r , 0], where t ∈ A ∈ Rd×d is a matrix, r is a constant delay, Cr := C ([−r , 0], Rd ) is the space of continuous functions on [−r , 0] valued in Rd , f and g are functions defined on Cr valued in Rd and Rd×m respectively, and Z is a Rm -valued stochastic process with stationary increments on
In memory of Russell Johnson.
B
Nguyen Dinh Cong [email protected] Luu Hoang Duc [email protected]; [email protected] Phan Thanh Hong [email protected]
1
Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam
2
Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany
3
Thang Long University, Hanoi, Vietnam
123
Journal of Dynamics and Differential Equations
a probability space (, F , P) which has almost sure all the realizations in the Hölder space C 0,ν for 21 < ν ≤ 1, the initial condition belongs to the Hölder space C 0,β0 ([−r , 0], Rd ). Equation (1.1) is understood in the path-wise sense using Young integration [25] for the stochastic term g(yt )d Z (t), whereas the term [Ay(t) + f (yt )]dt is defined by the classical Riemann-Stieltjes integration. For the notion of Young integral and its properties, as well as notions and properties of spaces of Hölder continuous functions and Hölder norms the reader is referred to Sect. 1 Appendix. In this paper, we investigate the asymptotic behavior of solution of the delay system (1.1) under regular assumptions. Namely, • H1 : A has all eigenvalues of negative real parts; • H2 : f is globally Lipschitz continuous and thus has linear growth, i.e there exists constants C f such that for all ξ, η ∈ Cr f (ξ ) − f (η) ≤ C f ξ − η∞,[−r ,0] ; • H3 : g is C 1 such that its Frechet derivative is bounded and locally Lipschitz continuous, i.e. there exists C g such that for all ξ, η ∈ Cr Dg(ξ ) L(Cr ,Rd ) ≤ C g , and for each M > 0, there exists L M such that for all ξ, η ∈ Cr satisfying ξ ∞,[
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