  We study the diffusion of a Brownian particle quadratically coupled to a
thermally fluctuating field. In the weak coupling limit, a path-integral
formulation allows to compute the effective diffusion coefficient in the cases
of an active particle, that tends to suppress the field fluctuations, and of a
passive particle, that only undergoes the field fluctuations. We show that the
behavior is similar to what was previously found for a linear coupling: an
active particle is always slowed down, whereas a passive particle is slowed
down in a slow field and accelerated in a fast field. Numerical simulations
show a good agreement with the analytical calculations. The examples of a
membrane protein coupled to the curvature or composition of the membrane are
discussed, with focus on the room for anomalous diffusion.
