Narrow escape with imperfect reactions

The imperfect narrow escape problem considers the mean first passage time (MFPT) of a Brownian particle through one of several small, partially reactive traps on an otherwise reflecting boundary within a confining domain. Mathematically, this problem is equivalent to Poisson's equation with mixed Neumann-Robin boundary conditions. Here, we obtain this MFPT in general three-dimensional domains by using strong localized perturbation theory in the small trap limit. These leading-order results involve factors, which are analogous to electrostatic capacitances, and we use Brownian local time theory and kinetic Monte Carlo (KMC) algorithms to rapidly compute these factors. Furthermore, we use a heuristic approximation of such a capacitance to obtain a simple, approximate MFPT, which is valid for any trap reactivity. In addition, we develop KMC algorithms to efficiently simulate the full problem and find excellent agreement with our asymptotic approximations.