  The problem of a particle diffusion in a fluctuating scalar field is studied.
In contrast to most studies of advection diffusion in random fields we analyze
the case where the particle position is also coupled to the dynamics of the
field. Physical realizations of this problem are numerous and range from the
diffusion of proteins in fluctuating membranes and the diffusion of localized
magnetic fields in spin systems. We present exact results for the diffusion
constant of particles diffusing in dynamical Gaussian fields in the adiabatic
limit where the field evolution is much faster than the particle diffusion. In
addition we compute the diffusion constant perturbatively, in the weak coupling
limit where the interaction of the particle with the field is small, using a
Kubo-type relation. Finally we construct a simple toy model which can be solved
exactly.
