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 is surrounded by semipermeable interface given closed surface \calM$ \subset \calM\subset \R^d$. That is, surrounds and thus partially screens process. also position of diffusing particle (searcher) randomly resets to initial $\x_0$ according Poisson process resetting rate $r$. The location taken be outside interface, $\x_0\in \calM^c$, which means does not occur when within interior \calM$. Hence, out effects resetting. first solve value problem (BVP) diffusion on half-line $x\in [0,\infty)$ an absorbing at $x=0$, barrier $x=L$, stochastic $x_0>L$ all $x>L$. calculate mean passage time (MFPT) absorbed explore behavior as function permeability $\kappa_0$ spatial $L$. then perform analogous calculations three-dimensional (3D) spherically symmetric target, show MFPT exhibits same qualitative 1D case. Finally, introduce single-particle realization based generalization so-called snapping BM. latter sews together successive rounds reflecting Brownian motion either side interface. main challenge establishing probability density generated BM satisfies permeable conditions how can achieved using renewal theory.