  We study the dynamics of a tracer in a dense mixture of particles connected
to different thermostats. Starting from the overdamped Langevin equations that
describe the evolution of the system, we derive the expression of the
self-diffusion coefficient of a tagged particle in the suspension, in the limit
of soft interactions between the particles. Our derivation, which relies on the
linearization of the Dean-Kawasaki equations obeyed by the density fields and
on a path-integral representation of the dynamics of the tracer, extends
previous derivations that held for tracers in contact with a single bath. Our
analytical result is confronted to results from Brownian dynamics simulations.
The agreement with numerical simulations is very good even for high densities.
We show how the diffusivity of tracers can be affected by the activity of a
dense environment of soft particles that may represent polymer coils -- a
result that could be of relevance in the interpretation of measurements of
diffusivity in biological media. Finally, our analytical result is general and
can be applied to the diffusion of tracers coupled to different types of
fluctuating environments, provided that their evolution equations are linear
and that the coupling between the tracer and the bath is weak.
