2204.04129
The Lyapunov Spectrum for Conditioned Random Dynamical Systems
Matheus M. Castro, Dennis Chemnitz, Hugo Chu, Maximilian Engel, Jeroen S. W. Lamb, Martin Rasmussen
correctmedium confidence
- Category
- Not specified
- Journal tier
- Strong Field
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves the existence of a full Lyapunov spectrum for the Q-process and L1-convergence of finite-time singular-value growth rates under Qν by invoking the multiplicative ergodic theorem (plus Furstenberg–Kesten) for a linear cocycle over the ergodic, invariant measure Qν. The candidate solution obtains precisely the same conclusion, but via a direct application of Kingman’s subadditive ergodic theorem to the exterior-power cocycles, together with a careful discrete-to-continuous-time reduction. The assumptions match the paper’s integrability and invertibility conditions, and the steps are logically sound. Hence both are correct, with different proof routes.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} strong field
\textbf{Justification:}
The paper develops a coherent and robust framework to prove the existence of a full conditioned Lyapunov spectrum for absorbed RDS by leveraging the Q-process and an ergodic, invariant base measure Qν. The main results are correct and significant, connecting quasi-stationary/ergodic theory with multiplicative ergodic techniques. Clarifying a few proof details—especially the L1 convergence step for individual singular values—and adding pointers to the subadditive exterior-power argument already used later would enhance readability.