Large deviations in the Langevin dynamics of a random field Ising model

We consider a Langevin dynamics scheme for a d-dimensional Ising model with a disordered external magnetic field and establish that the averaged law of the empirical process obeys a large deviation principle (LDP), according to a good rate functional Ja having a unique minimiser Q∞. The asymptotic dynamics Q∞ may be viewed as the unique weak solution associated with an infinite-dimensional system of interacting diffusions, as well as the unique Gibbs measure corresponding to an interaction Ψ on infinite dimensional path space. We then show that the quenched law of the empirical process also obeys a LDP, according to a deterministic good rate functional Jq satisfying: Jq⩾Ja, so that (for a typical realisation of the disordered external magnetic field) the quenched law of the empirical process converges exponentially fast to a Dirac mass concentrated at Q∞.