  We present the general solutions for the classical and quantum dynamics of
the anharmonic oscillator coupled to a purely diffusive environment. In both
cases, these solutions are obtained by the application of the
Baker-Campbell-Hausdorff (BCH) formulas to expand the evolution operator in an
ordered product of exponentials. Moreover, we obtain an expression for the
Wigner function in the quantum version of the problem. We observe that the role
played by diffusion is to reduce or to attenuate the the characteristic quantum
effects yielded by the nonlinearity, as the appearance of coherent
superpositions of quantum states (Schr\"{o}dinger cat states) and revivals.
