  We investigate how confinement may drastically change both the probability
density of the first-encounter time and the related survival probability in the
case of two diffusing particles. To obtain analytical insights into this
problem, we focus on two one-dimensional settings: a half-line and an interval.
We first consider the case with equal particle diffusivities, for which exact
results can be obtained for the survival probability and the associated
first-encounter time density over the full time domain. We also evaluate the
moments of the first-encounter time when they exist. We then turn to the case
when the diffusivities are not equal, and focus on the long-time behavior of
the survival probability. Our results highlight the great impact of boundary
effects in diffusion-controlled kinetics even for simple one-dimensional
settings, as well as the difficulty of obtaining analytic results as soon as
translational invariance of such systems is broken.
