On the gap between deterministic and probabilistic Lyapunov exponents for continuous-time linear systems

TL;DR

This paper investigates the relationship between deterministic and probabilistic Lyapunov exponents in continuous-time linear systems with piecewise constant matrices, establishing convergence results for associated Markov processes.

Contribution

It provides new insights into when the maximal Lyapunov exponents coincide and proves a convergence theorem for Markov processes linked to these systems.

Findings

Conditions for equality of Lyapunov exponents identified.

Markov processes converge to those associated with convex combinations of matrices.

New convergence theorem for Markov processes in this context.

Abstract

Consider a non-autonomous continuous-time linear system in which the time-dependent matrix determining the dynamics is piecewise constant and takes finitely many values $A_1, \dotsc, A_N$. This paper studies the equality cases between the maximal Lyapunov exponent associated with the set of matrices $\{A_1, \dotsc, A_N\}$, on the one hand, and the corresponding ones for piecewise deterministic Markov processes with modes $A_1, \dotsc, A_N$, on the other hand. A fundamental step in this study consists in establishing a result of independent interest, namely, that any sequence of Markov processes associated with the matrices $A_1,\dotsc, A_N$ converges, up to extracting a subsequence, to a Markov process associated with a suitable convex combination of those matrices.