  We consider the problem of determining the arrival statistics of unbiased
planar random walkers to complex target configurations. In contrast to problems
posed in finite domains, simple moments of the distribution, such as the mean
(MFPT) and variance, are not defined and it is necessary to obtain the full
arrival statistics. We describe several methods to obtain these distributions
and other associated quantities such as splitting probabilities. One approach
combines a Laplace transform of the underlying parabolic equation with matched
asymptotic analysis followed by numerical transform inversion. The second
approach is similar, but uses a boundary integral equation method to solve for
the Laplace transformed variable. To validate the results of this theory, and
to obtain the arrival time statistics in very general configurations of
absorbers, we introduce an efficient Kinetic Monte Carlo (KMC) method that
describes trajectories as a combination of large but exactly solvable
projection steps. The effectiveness of these methodologies is demonstrated on a
variety of challenging examples highlighting the applicability of these methods
to a variety of practical scenarios, such as source inference. A particularly
useful finding arising from these results is that homogenization theories, in
which complex configurations are replaced by equivalent simple ones, are
remarkably effective at describing arrival time statistics.
