Linear systems with adiabatic fluctuations

We consider a dynamical system subjected to weak but adiabatically slow fluctuations of external origin. Based on the ``adiabatic following'' approximation we carry out an expansion in \alpha/|\mu|, where \alpha is the strength of fluctuations and 1/|\mu| refers to the time scale of evolution of the unperturbed system to obtain a linear differential equation for the average solution. The theory is applied to the problems of a damped harmonic oscillator and diffusion in a turbulent fluid. The result is the realization of `renormalized' diffusion constant or damping constant for the respective problems. The applicability of the method has been critically analyzed.