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

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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.
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Dates and versions

hal-03478271 , version 1 (13-12-2021)
hal-03478271 , version 2 (21-11-2022)

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Yacine Chitour, Guilherme Mazanti, Pierre Monmarché, Mario Sigalotti. On the gap between deterministic and probabilistic Lyapunov exponents for continuous-time linear systems. 2022. ⟨hal-03478271v2⟩
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