  The motion of a colloidal probe in a complex fluid, such as a micellar
solution, is usually described by the generalized Langevin equation, which is
linear. However, recent numerical simulations and experiments have shown that
this linear model fails when the probe is confined, and that the intrinsic
dynamics of the probe is actually non-linear. Noting that the kurtosis of the
displacement of the probe may reveal the non-linearity of its dynamics also in
the absence confinement, we compute it for a probe coupled to a Gaussian field
and possibly trapped by a harmonic potential. We show that the excess kurtosis
increases from zero at short times, reaches a maximum, and then decays
algebraically at long times, with an exponent which depends on the spatial
dimensionality and on the features and correlations of the dynamics of the
field. Our analytical predictions are confirmed by numerical simulations of the
stochastic dynamics of the probe and the field where the latter is represented
by a finite number of modes.
