The evolution of a scalar field interacting with an environment in the de Sitter phase of an inflationary Universe is studied. The environment is taken to be a second scalar field. It is shown that the coherence length of the quantum fluctuations rapidly decreases after the wavelength of the perturbation crosses the Hubble radius. Hence, the fluctuations can be interpreted as classical. This lends support ot the usual derivation of the spectrum of density perturbations in inflationary Universe models.