title: The escape problem for mortal walkers.
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. Our findings provide a mathematical ground for optimizing storage containers and materials to reduce the risk of leakage of dangerous chemicals or nuclear wastes.
authors.affiliation: Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS - Ecole Polytechnique, University Paris-Saclay, 91128 Palaiseau, France.
authors.forename: D S
authors.identifier: 0000000262739164
authors.initials: DS
authors.lastname: Grebenkov
authors.affiliation: Mechanobiology Institute, National University of Singapore, 5A Engineering Drive 1, Singapore 117411, Singapore.
authors.forename: J-F
authors.identifier: 0000000189045878
authors.initials: JF
authors.lastname: Rupprecht
mesh_terms: D000818:Animals; D016923:Cell Death; D003198:Computer Simulation; D004058:Diffusion; D005260:Female; D005306:Fertilization; D005456:Fluorescent Dyes / Q000737:chemistry; D006801:Humans; D007700:Kinetics; D008297:Male; D008954:Models, Biological; D008956:Models, Chemical; D008958:Models, Molecular; D038761:Photobleaching; D011336:Probability; D020871:RNA Stability; D012333:RNA, Messenger / Q000737:chemistry; D013094:Spermatozoa / Q000166:cytology; D013269:Stochastic Processes
chemical_list: D005456:Fluorescent Dyes; D012333:RNA, Messenger