| Abstract: |
| I will describe work in a program to extract statistical properties of random dynamical systems using spectral approaches based on transfer operator cocycles and Lyapunov exponents. A random dynamical system is governed by a cocycle of nonlinear transformations on a state space and this naturally induces a cocycle of linear transfer operators acting on suitable Banach spaces. By using random perturbation theory to study derivatives of Lyapunov exponents of ``twisted`` transfer operator cocycles, we obtain elegant proofs of statistical limit laws such as a central limit theorem. Inserting random holes into the phase space, we produce a nonstationary extreme value theory for a random observation function, whereby observing an extreme value corresponds to landing in a random hole and terminating the trajectory. Considering the holes as ``targets``, we further extend this formalism to count the number of visits to random targets in a random orbit, thereby capturing the counting distribution of extreme-value occurrences in random trajectories of increasing length. All results are in the ``quenched`` sense, meaning that they hold almost surely across all random realisations of the dynamics and observations. |
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