Asymptotic analysis of extended two-dimensional narrow capture problems

Abstract

In this paper we extend our recent work on two-dimensional (2D) diffusive search-and-capture processes with multiple small targets (narrow capture problems) by considering an asymptotic expansion of the Laplace transformed probability flux into each target. The latter determines the distribution of arrival or capture times into an individual target, conditioned on the set of events that result in capture by that target. A characteristic feature of strongly localized perturbations in 2D is that matched asymptotics generates a series expansion in $\nu =-1/\ln \epsilon$ rather than $\epsilon$, $0<\epsilon \ll 1$, where $\epsilon$ specifies the size of each target relative to the size of the search domain. Moreover, it is possible to sum over all logarithmic terms non-perturbatively. We exploit this fact to show how a Taylor expansion in the Laplace variable $s$ for fixed $\nu$ provides an efficient method for obtaining corresponding asymptotic expansions of the splitting probabilities and moments of the conditional FPT densities. We then use our asymptotic analysis to derive new results for two major extensions of the classical narrow capture problem: optimal search strategies under stochastic resetting, and the accumulation of target resources under multiple rounds of search-and-capture.

22 pages, 5 figures

References in corpus (6)

• Stochastic Resetting and Applications
• First Passage Under Restart
• First passage under stochastic resetting in an interval
• Search with home returns provides advantage under high uncertainty
• Resetting processes with non-instanteneous return
• Hybrid asymptotic-numerical approach for estimating first passage time densities of the two-dimensional narrow capture problem

Cited by in corpus (4)

• The inspection paradox in stochastic resetting
• The narrow capture problem: an encounter-based approach to partially reactive targets
• Accumulation time of diffusion in a 3D singularly perturbed domain
• 2D interfacial diffusion model of inhibitory synaptic receptor dynamics