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We consider quasi-compact linear operator cocycles$\mathcal{L}^{n}_\omega:=\mathcal{L}_{\sigma^{n-1}\omega}\circ\cdots\circ\mathcal{L}_{\sigma\omega}\circ\mathcal{L}_{\omega}$ driven by an invertible ergodic process$\sigma:\Omega\to\Omega$, and their small perturbations$\mathcal{L}_{\omega,\epsilon}^{n}$. We prove an abstract $\omega$-wisefirst-order formula for the leading Lyapunov multipliers. We then consider thesituation where $\mathcal{L}_{\omega}^{n}$ is a transfer operator cocycle for arandom map cocycle $T_\omega^{n}:=T_{\sigma^{n-1}\omega}\circ\cdots\circT_{\sigma\omega}\circ T_{\omega}$ and the perturbed transfer operators$\mathcal{L}_{\omega,\epsilon}$ are defined by the introduction of small randomholes $H_{\omega,\epsilon}$ in $[0,1]$, creating a random open dynamicalsystem. We obtain a first-order perturbation formula in this setting, whichreads$\lambda_{\omega,\epsilon}=\lambda_\omega-\theta_{\omega}\mu_{\omega}(H_{\omega,\epsilon})+o(\mu_\omega(H_{\omega,\epsilon})),$where $\mu_\omega$ is the unique equivariant random measure (and equilibriumstate) for the original closed random dynamics. Our new machinery is thendeployed to create a spectral approach for a quenched extreme value theory thatconsiders random dynamics with general ergodic invertible driving, and randomobservations. An extreme value law is derived using the first-order terms$\theta_\omega$. Further, in the setting of random piecewise expanding intervalmaps, we establish the existence of random equilibrium states and conditionallyinvariant measures for random open systems via a random perturbative approach.Finally we prove quenched statistical limit theorems for random equilibriumstates arising from contracting potentials. We illustrate the theory with avariety of explicit examples.

Perturbation formulae for quenched random dynamics with applications to open systems and extreme value theory