  In this work we study fluctuations and dissipation of a string in a deformed
anti-de Sitter (AdS) space at finite temperature and density. The deformed AdS
space is a charged black hole solution of the Einstein-Maxwell-Dilaton action.
In this background we take into account the backreaction on the horizon
function from an exponential deformation of the AdS space. In this set up, we
study the equations of motion of the string from the Nambu-Goto action. The
motion of the string endpoint describes holographically the fluctuations and
dissipation of a particle moving in a four-dimensional Minkowski space. From
this model we compute the admittance and study its behavior against the
temperature for some values of the chemical potential. In order to obtain the
two-point correlations functions, and the mean square displacement we consider
separately the bosonic and fermionic cases with $\mu < 0$ and $\mu > 0$,
respectively. From the regularized mean square displacements of the bosonic and
fermionic cases we obtain the short and large time approximations. For the
short time case we find the usual ballistic regime, and for the long time case
we obtain a sub-diffusive regime characteristic of a ultra-slow scaled Brownian
motion. Combining the results from the admittance and mean square displacements
we also study the fluctuation-dissipation theorem.
