Effect of Partial Absorption on Diffusion with Resetting

The effect of partial absorption on a diffusive particle which stochastically resets its position with a finite rate $r$ is considered. The particle is absorbed by a target at the origin with absorption `velocity' $a$; as the velocity $a$ approaches $\infty$ the absorption property of the target approaches that of a perfectly-absorbing target. The effect of partial absorption on first-passage time problems is studied, in particular, it is shown that the mean time to absorption (MTA) is increased by an additive term proportional to $1/a$. The results are extended to multiparticle systems where independent searchers, initially uniformly distributed with a given density, look for a single immobile target. It is found that the average survival probability $P^{av}$ is modified by a multiplicative factor which is a function of $1/a$, whereas the decay rate of the typical survival probability $P^{typ}$ is decreased by an additive term proportional to $1/a$.