Abstract. 
We consider Markov processes that induce a random dynamical system 
evolving in a domain with forbidden states constituting a trap.
We investigate the dynamical behavior of the process before hitting the trap, 
asking what happens when one conditions the process to survive for a long 
time. Using concepts like quasi-stationary and quasi-ergodic distributions, 
we can define average Lyapunov exponents and describe the bifurcation 
behavior of typical examples of stochastic bifurcation theory within this environment. 
The underlying philosophy is to exhibit the local character of 
random bifurcations for stochastic differential equations which are usually 
hidden in the global analysis. If time permits, we'll also discuss relations to 
dynamical systems with holes.