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 $\beta$, 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 $D_\text{eff}$, which is induced by the trapping environment and is typically different from the original one. We show that when the number of traps is \emph{finite}, the environment enhances diffusion and induces an effective diffusion coefficient that is systematically equal to $D_\text{eff}=2D$, independently of the number of the traps, the trapping intensity $\beta$ and the distance $L$. On the contrary, when the number of traps is \emph{infinite}, we find that the environment inhibits diffusion with an effective diffusion coefficient that depends on the traps intensity $\beta$ and the distance $L$ through a non-trivial scaling function $D_\text{eff}=D \mathcal{F}(\beta L/D)$, 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.