Diffusion with stochastic resetting screened by a semipermeable interface

In this paper we consider the diffusive search for a bounded target Omega is an element of R-d with its boundary partial derivative Omega totally absorbing. We assume that the target is surrounded by a semipermeable interface given by the closed surface partial derivative M with Omega subset of M subset of R-d. That is, the interface totally surrounds the target and thus partially screens the diffusive search process. We also assume that the position of the diffusing particle (searcher) randomly resets to its initial position x0 according to a Poisson process with a resetting rate r. The location x(0) is taken to be outside the interface, x(0) is an element of M-c, which means that resetting does not occur when the particle is within the interior of partial derivative M. (Otherwise, the particle would have to cross the interface in order to reset to x(0).) Hence, the semipermeable interface also screens out the effects of resetting. We illustrate the theory by explicitly solving the boundary value problem for a three-dimensional (3D) spherically symmetric interface and a concentric spherical target. We calculate the mean first passage time (MFPT) to find (be absorbed by) the target and explore its behavior as a function of the permeability kappa(0) of the interface and the radius of the interface R. In particular, we find that increasing R for a fixed target size reduces the MFPT and increases the optimal resetting rate at which the MFPT is minimized. We also find that the sensitivity of the MFPT to changes in kappa(0) is a decreasing function of R. Finally, we introduce a stochastic single-particle realization of the search process based on a generalization of so-called snap-ping out Brownian motion (BM). The latter sews together successive rounds of reflecting BM on either side of the interface. The main challenge is establishing that the probability density generated by the snapping out BM satisfies the permeable boundary conditions at the interface. We show how this can be achieved using renewal theory.