  We investigate a random integral which provides a natural example of an
imaginary exponential functional of Brownian motion. This functional shows up
in the study of the binary annihilation process, within the Doi-Peliti
formalism for reaction-diffusion systems. The main emphasis is put on the
complementarity between the usual Langevin approach and another approach based
on the similarity with Kesten variables and other one-dimensional disordered
systems. Even though neither of these routes leads to the full solution of the
problem, we have obtained a collection of results describing various regimes of
interest.
