Exact identities are derived for a family of models including (a) a domain wall in a random field Ising model (RFIM), and (b)the random anisotropy XY model in the no-vortex approximation. In particular, the second moment of thermal fluctuations is not affected by frozen randomness. It is checked in a one- dimensional model that higher moments are on the contrary strongly enhanced. Thus, thermal fluctuations are strongly non-Gaussian. This reflects excursions between remote potential wells in the phase space. It is shown exactly that the Imry-Ma argument yields a correct evaluation of the field-induced fluctuations for the one-dimensional model.