  Light vector fields during inflation obtain a superhorizon perturbation
spectrum when their conformal invariance is appropriately broken. Such
perturbations, by means of some suitable mechanism (e.g. the vector curvaton
mechanism), can contribute to the curvature perturbation in the Universe and
produce characteristic signals, such as statistical anisotropy, on the
microwave sky, most recently surveyed by the Planck satellite mission. The
magnitude of such characteristic features crucially depends on the magnitude of
the vector condensate generated during inflation. However, the expectation
value of this condensate has so-far been taken as a free parameter, lacking a
definite prediction or a physically motivated estimate. In this paper, we study
the stochastic evolution of the vector condensate and obtain an estimate for
its magnitude. Our study is mainly focused in the supergravity inspired case
when the kinetic function and mass of the vector boson is time-varying during
inflation, but other cases are also explored such as a parity violating axial
theory or a non-minimal coupling between the vector field and gravity. As an
example, we apply our findings in the context of the vector curvaton mechanism
and contrast our results with current observations.
