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

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 $A_1,\dots ,A_N$. This paper studies the equality cases between the maximal Lyapunov exponent associated with the set of matrices $\{A_1,\dots ,A_N\}$, on the one hand, and the corresponding ones for piecewise deterministic Markov processes with modes $A_1,\dots ,A_N$, 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 $A_1,\dots ,A_N$ converges, up to extracting a subsequence, to a Markov process associated with a suitable convex combination of those matrices.

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• Metastable Markov chains: from the convergence of the trace to the convergence of the finite-dimensional distributions
• Dynamical systems with fast switching and slow diffusion: Hyperbolic equilibria and stable limit cycles
• On the gap between deterministic and probabilistic joint spectral radii for discrete-time linear systems