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. This is also a mathematical ground for optimizing storage containers and materials to reduce the risk of leakage of dangerous chemicals or nuclear wastes.

References in corpus (18)

• Optimal reaction time for surface-mediated diffusion
• Mortality, Redundancy, and Diversity in Stochastic Search
• Kinetics of stochastically-gated diffusion-limited reactions and geometry of random walk trajectories
• Reaction-subdiffusion and reaction-superdiffusion equations for evanescent particles performing continuous time random walks
• Universal formula for the mean first passage time in planar domains
• Diffusive escape through a narrow opening: new insights into a classic problem
• Exploration and Trapping of Mortal Random Walkers
• Mean first-passage time of surface-mediated diffusion in spherical domains
• The subdiffusive target problem: Survival probability
• Exit time distribution in spherically symmetric two-dimensional domains
• Partially Reflected Brownian Motion: A Stochastic Approach to Transport Phenomena
• Kinetics of active surface-mediated diffusion in spherically symmetric domains
• Exact mean exit time for surface-mediated diffusion
• Survival probability of an immobile target in a sea of evanescent diffusive or subdiffusive traps: a fractional equation approach
• Uniform asymptotic approximation of diffusion to a small target
• Evanescent continuous time random walks
• Narrow-escape-time problem: the imperfect trapping case
• The number statistics and optimal history of non-equilibrium steady states of mortal diffusing particles

Cited by in corpus (32)

• Random walks and diffusion on networks
• Restart could optimize the probability of success in a Bernoulli trial
• Paradigm shift in diffusion-mediated surface phenomena
• Towards a full quantitative description of single-molecule reaction kinetics in biological cells
• Distribution of extreme first passage times of diffusion
• From single-particle stochastic kinetics to macroscopic reaction rates: fastest first-passage time of $N$ random walkers
• Spectral theory of imperfect diffusion-controlled reactions on heterogeneous catalytic surfaces
• Narrow escape of interacting diffusing particles
• Probability distribution of the boundary local time of reflected Brownian motion in Euclidean domains
• Surface Hopping Propagator: An Alternative Approach to Diffusion-Influenced Reactions
• Heterogeneous continuous time random walks
• Diffusion toward non-overlapping partially reactive spherical traps: fresh insights onto classic problems
• Preface: New trends in first-passage methods and applications in the life sciences and engineering
• Joint distribution of multiple boundary local times and related first-passage time problems with multiple targets
• Depletion of Resources by a Population of Diffusing Species
• Statistics of diffusive encounters with a small target: Three complementary approaches
• Distribution of first-reaction times with target regions on boundaries of shell-like domains
• Escape of a Sticky Particle
• An encounter-based approach to the escape problem
• Motion of a Brownian molecule in the presence of reactive boundaries
• First-passage properties of mortal random walks: ballistic behavior, effective reduction of dimensionality, and scaling functions for hierarchical graphs
• Mortal Brownian motion: three short stories
• Effects of mortality on stochastic search processes with resetting
• First-encounter time of two diffusing particles in two- and three-dimensional confinement
• Diffusion towards a nanoforest of absorbing pillars
• Target finding in fibrous biological environments
• A molecular relay race: sequential first-passage events to the terminal reaction centre in a cascade of diffusion controlled processes
• Survival in a nanoforest of absorbing pillars
• Escape-from-a-layer approach for simulating the boundary local time in Euclidean domains
• Random Walks on Lattices. Influence of Competing Reaction Centers on Diffusion-Controlled Processes
• Adsorption and permeation events in molecular diffusion
• An encounter-based approach for restricted diffusion with a gradient drift