  We study a Thouless pump realized with an elastically \textit{deformable
quantum dot} whose center of mass follows a non-linear stochastic dynamics. The
interplay of noise, non-linear effects, dissipation and interaction with an
external time-dependent driving on the pumped charge is fully analyzed. The
results show that the quantum pumping mechanism not only is not destroyed by
the force fluctuations, but it becomes stronger when the forcing signal
frequency is tuned close to the resonance of the vibrational mode. The
robustness of the quantum pump with temperature is also investigated and an
exponential decay of the pumped charge is found when the coupling to the
vibrational mode is present. Implications of our results for
nano-electromechanical systems are also discussed.
