We consider a particle evolving according to a Markov motion in an absorbing medium. We analyze the long term behavior of the time at which the particle is killed and the distribution of the particle conditional upon survival. Under given regularity conditions, these quantities are characterized by the limiting distribution and the Lyapunov exponent of a nonlinear Feynman-Kac operator. We propose to approximate numerically this distribution and this exponent based on various interacting particle system interpretations of the Feynman-Kac operator. We study the properties of the resulting estimates.