Nonautonomous Young Differential Equations Revisited

Abstract

In this paper we prove that under mild conditions a nonautonomous Young differential equation possesses a unique solution which depends continuously on initial conditions. The proofs use estimates in p-variation norms, the construction of greedy sequence of times, and Gronwall-type lemma with the help of Shauder theorem of fixed points.

Lyapunov Spectrum of Nonautonomous Linear Young Differential Equations

“…Then τ k → ∞ as k → ∞ (see the proof in [9]). Denote by N (a, b, ω) the number of τ k in the finite interval (a, b], then from [9]…”

“…AppendixProof: [Proposition 2.2] The proof follows the same techniques in[25] and in[9] with some modifications. First, consider x ∈ C q−var ([a, b], R d ) with some [a, b] ⊂ [t 0 , t 0 +T ].…”

Asymptotic Dynamics of Young Differential Equations

“…where B H i are scalar fractional Brownian motions (not necessarily independent), we can apply Lemma 2.1 in [6] with the estimate in [9, Lemma 4.1 (iii), p.14] to obtain that ( p) < ∞. Proposition 3.…”

“…Here we would like to construct, for any γ > 0 and any given interval [a, b], a sequence of greedy times {τ k (γ )} k∈N as follows (see e.g. [5,6,8])…”

“…The existence and uniqueness theorem for Young differential equations is proved in many versions, e.g. [6,[17][18][19]21,25].…”