Lyapunov spectrum of nonautonomous linear Young differential equations

TL;DR

This paper establishes the Lyapunov spectrum for nonautonomous linear Young differential equations, showing its independence from the driving path in certain systems and analyzing the stochastic case.

Contribution

It introduces a framework for defining and computing Lyapunov exponents for Young differential equations, including stochastic systems, and proves generic regularity results.

Findings

Lyapunov spectrum is well-defined for these equations.

Spectrum can be computed via discretized flow.

Almost all such equations are Lyapunov regular.

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.