Lyapunov Spectrum of Nonautonomous Linear Young Differential Equations
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Lyapunov Spectrum of Nonautonomous Linear Young Differential Equations Nguyen Dinh Cong1 · Luu Hoang Duc1,2 · Phan Thanh Hong3 In memory of V. M. Millionshchikov Received: 31 January 2019 / Revised: 3 July 2019 © The Author(s) 2019
Abstract We show that a linear Young differential equation generates a topological two-parameter flow, thus the notions of Lyapunov exponents and Lyapunov spectrum are well-defined. The spectrum can be computed using the discretized flow and is independent of the driving path for triangular systems which are regular in the sense of Lyapunov. In the stochastic setting, the system generates a stochastic two-parameter flow which satisfies the integrability condition, hence the Lyapunov exponents are random variables of finite moments. Finally, we prove a Millionshchikov theorem stating that almost all, in a sense of an invariant measure, linear nonautonomous Young differential equations are Lyapunov regular. Keywords Young differential equation · Two parameter flow · Lyapunov exponent · Lyapunov spectrum · Lyapunov regularity · Multiplicative ergodic theorem · Bebutov flow
1 Introduction In this article we study the Lyapunov spectrum of the nonautonomous linear Young differential equation (abbreviated by YDE) d x(t) = A(t)x(t)dt + C(t)x(t)dω(t), x(t0 ) = x0 ∈ Rd , t ≥ t0 ,
(1.1)
where A, C are continuous matrix valued functions on [0, ∞), and ω is a continuous path on [0, ∞) having finite p-th variation on each compact interval of [0, ∞), for some p ∈ (1, 2).
B
Luu Hoang Duc [email protected]; [email protected] Nguyen Dinh Cong [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
Such system (1.1) appears, for instance, when considering the linearization of the autonomous Young differential equation dy(t) = f (y(t))dt + g(y(t))dω(t)
(1.2)
along any reference solution y(t, y0 , ω). An example is when we would like to solve in the pathwise sense stochastic differential equations driven by fractional Brownian motions with Hurst index H ∈ ( 21 , 1) defined on a complete probability space (, F , P) [24]. In fact it follows from [5] that (1.2) under the stochastic setting also satisfies the integrability condition. The Eq. (1.1) can be rewritten in the integral form t t x(t) = x0 + A(s)x(s)ds + C(s)x(s)dω(s), t ≥ t0 , (1.3) t0
t0
where the second integral is understood in the Young sense [28], which can also be presented in terms of fractional derivatives [29]. Under some mild conditions, the unique solution of (1.1) generates a two-parameter flow ω (t0 , t), as seen in [9]. Under a certain stochastic setting, (1.1) actually generates a stochastic two-parameter flow in the sense of Kunita [16]. Our aim is to study the Lyapunov exponents and Lyapunov spectrum of the linear twoparameter flow generated from Young E
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