  We develop a framework for the stochastic thermodynamics of a probe coupled
to a fluctuating medium with spatio-temporal correlations, described by a
scalar field. For a Brownian particle dragged by a harmonic trap through a
fluctuating Gaussian field, we show that near criticality (where the field
displays long-range spatial correlations) the spatially-resolved average heat
flux develops a dipolar structure, where heat is absorbed in front and
dissipated behind the dragged particle. Moreover, a perturbative calculation
reveals that the dissipated power displays three distinct dynamical regimes
depending on the drag velocity.
