  We consider the problem of building a continuous stochastic model, i.e. a
Langevin or Fokker-Planck equation, through a well-controlled coarse-graining
procedure. Such a method usually involves the elimination of the fast degrees
of freedom of the "bath" to which the particle is coupled. Specifically, we
look into the general case where the bath may be at negative temperatures, as
found - for instance - in models and experiments with bounded effective kinetic
energy. Here, we generalise previous studies by considering the case in which
the coarse-graining leads to (i) a renormalisation of the potential felt by the
particle, and (ii) spatially dependent viscosity and diffusivity. In addition,
a particular relevant example is provided, where the bath is a spin system and
a sort of phase transition takes place when going from positive to negative
temperatures. A Chapman-Enskog-like expansion allows us to rigorously derive
the Fokker-Planck equation from the microscopic dynamics. Our theoretical
predictions show an excellent agrement with numerical simulations.
