  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.
