Gradual Diffusive Capture: Slow Death by Many Mosquito Bites
S. Redner, O. B\'enichou

TL;DR
This paper models a diffusing particle repeatedly attacked by another diffusing particle, where each attack reduces the former's mobility until it 'dies', revealing a novel stochastic process with specific displacement and time distributions.
Contribution
It introduces a new model of gradual diffusive capture with decreasing diffusivity, analyzing the resulting displacement and lifetime distributions of the 'man' particle.
Findings
Displacement distribution follows a Cauchy form decaying as x^{-2}.
Time until death distribution decays as t^{-3/2}.
Man's diffusivity reaches zero after multiple attacks.
Abstract
We study the dynamics of a single diffusing particle (a "man") with diffusivity that is attacked by another diffusing particle (a "mosquito") with fixed diffusivity . Each time the mosquito meets and bites the man, the diffusivity of the man is reduced by a fixed amount, while the diffusivity of the mosquito is unchanged. The mosquito is also displaced by a small distance with respect to the man after each encounter. The man is defined as dead when reaches zero. At the moment when the man dies, his probability distribution of displacements is given by a Cauchy form, which asymptotically decays as , while the distribution of times when the man dies asymptotically decays as , which has the same form as the one-dimensional first-passage probability.
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