The escape problem for mortal walkers

TL;DR

This paper studies the escape problem for mortal random walkers, analyzing how death rates affect survival, escape probabilities, and passage times, with applications in bio-imaging, genetics, fertilization, and safety.

Contribution

It introduces a framework for the escape problem with mortality, deriving bounds, asymptotics, and solutions for various regimes, expanding understanding of stochastic processes with decay.

Findings

Identified three asymptotic regimes of death rates.

Derived bounds and approximations for survival and escape probabilities.

Validated results with explicit solutions and numerical simulations.

Abstract

We introduce and investigate the escape problem for random walkers that may eventually die, decay, bleach, or lose activity during their diffusion towards an escape or reactive region on the boundary of a confining domain. In the case of a first-order kinetics (i.e., exponentially distributed lifetimes), we study the effect of the associated death rate onto the survival probability, the exit probability, and the mean first passage time. We derive the upper and lower bounds and some approximations for these quantities. We reveal three asymptotic regimes of small, intermediate and large death rates. General estimates and asymptotics are compared to several explicit solutions for simple domains, and to numerical simulations. These results allow one to account for stochastic photobleaching of fluorescent tracers in bio-imaging, degradation of mRNA molecules in genetic translation mechanisms, or high mortality rates of spermatozoa in the fertilization process. This is also a mathematical ground for optimizing storage containers and materials to reduce the risk of leakage of dangerous chemicals or nuclear wastes.