Conditioned Lyapunov exponents for random dynamical systems

We introduce the notion of conditioned Lyapunov exponents for random dynamical systems, where we condition on trajectories that stay within a bounded domain for asymptotically long times. This is motivated by the desire to characterise local dynamical properties in the presence of unbounded noise (when almost all trajectories are unbounded). We present two different approaches to prove the existence of such conditioned Lyapunov exponents, and both approaches make use of recent developments on quasi-stationary and quasi-ergodic measures. (joint work with Matheus de Castro, Dennis Chemnitz, Hugo Chu, Maximilian Engel, and Jeroen Lamb)