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 ∞ 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.