Transport properties of diffusive particles conditioned to survive in trapping environments

We consider a one-dimensional Brownian motion with diffusion coefficient D in the presence of n partially absorbing traps with intensity β, separated by a distance L and evenly spaced around the initial position of the particle. We study the transport properties of the process conditioned to survive up to time t. We find that the surviving particle first diffuses normally, before it encounters the traps, then undergoes a period of transient anomalous diffusion, after which it reaches a final diffusive regime. The asymptotic regime is governed by an effective diffusion coefficient D eff, which is induced by the trapping environment and is typically different from the original one. We show that when the number of traps is finite, the environment enhances diffusion and induces an effective diffusion coefficient that is systematically equal to D eff = 2D, independently of the number of the traps, the trapping intensity β and the distance L. On the contrary, when the number of traps is infinite, we find that the environment inhibits diffusion with an effective diffusion coefficient that depends on the traps intensity β and the distance L through a non-trivial scaling function D eff = D F ( β L / D ) , for which we obtain a closed-form. Moreover, we provide a rejection-free algorithm to generate surviving trajectories by deriving an effective Langevin equation with an effective repulsive potential induced by the traps. Finally, we extend our results to other trapping environments.