  We describe a tracer in a bath of soft Brownian colloids by a particle
coupled to the density field of the other bath particles. From the Dean
equation, we derive an exact equation for the evolution of the whole system,
and show that the density field evolution can be linearized in the limit of a
dense bath. This linearized Dean equation with a tracer taken apart is
validated by the reproduction of previous results on the mean-field liquid
structure and transport properties. Then, the tracer is submitted to an
external force and we compute the density profile around it, its mobility and
its diffusion coefficient. Our results exhibit effects such as bias enhanced
diffusion that are very similar to those observed in the opposite limit of a
hard core lattice gas, indicating the robustness of these effects. Our
predictions are successfully tested against molecular dynamics simulations.
