We study the statistical properties of open random dynamical systems via absorbing Markov processes. This thesis is structured \add{into} two parts. The first \add{part} addresses the existence of quasi-ergodic measures, \add{and the second part} establishes \add{the existence of} conditioned Lyapunov exponents \add{under} suitable conditions.

In Chapters \ref{chapter:SF} and \ref{chapter:F}, leveraging the Banach lattice theory, we establish the existence of quasi-stationary and quasi-ergodic measures via a functional analytic \add{approach} for a wide class of open random dynamical systems, \add{under mild} assumptions on their transition kernel and transitiv\add{ity} properties, \add{p}roposing a novel and effective approach to the existence of such measures.

In Chapter \ref{Chapter:CLE}, we use the objects established in the previous chapters to prove the existence of a fully conditioned Lyapunov spectrum for open random dynamical systems. Also, we propose a method to treat smooth, conditioned dynamics effectively and robustly, providing a novel perspective on their long-term behaviour before escape.