We study Brownian motion in a Poissonian field of soft traps, conditioned on survival up to a given time horizon. The limit of this conditioned process, as the time horizon tends to infinity, is a diffusion known as the Q-process. On hyperbolic space, we use the spectral gap of the Laplacian to prove the existence of this limiting diffusion and study its properties. The main tool is to view the environment from the Brownian particle, turning the motion into dynamics on the foliated space associated with the random environment. This is joint work with Miklós Abért and Jaelin Kim, arXiv: 2602.18433.