Diffusion with stochastic resetting screened by a semipermeable interface

In this paper we consider the diffusive search for a bounded target $\Omega \in \R^d$ with its boundary $\partial \Omega$ totally absorbing. We assume that the target is surrounded by a semipermeable interface given by the closed surface $\partial \calM$ with $\Omega \subset \calM\subset \R^d$. 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 $\x_0$ according to a Poisson process with a resetting rate $r$. The location $\x_0$ is taken to be outside the interface, $\x_0\in \calM^c$, which means that resetting does not occur when the particle is within the interior of $\partial \calM$. Hence, the semipermeable interface also screens out the effects of resetting. We first solve the boundary value problem (BVP) for diffusion on the half-line $x\in [0,\infty)$ with an absorbing boundary at $x=0$, a semipermeable barrier at $x=L$, and stochastic resetting to $x_0>L$ for all $x>L$. We calculate the mean first passage time (MFPT) to be absorbed by the target and explore its behavior as a function of the permeability $\kappa_0$ of the interface and its spatial position $L$. We then perform the analogous calculations for a three-dimensional (3D) spherically symmetric interface and target, and show that the MFPT exhibits the same qualitative behavior as the 1D case. Finally, we introduce a stochastic single-particle realization of the search process based on a generalization of so-called snapping out BM. The latter sews together successive rounds of reflecting Brownian motion 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.