  In a general class of one dimensional random differential equation the
convergence of the distribution function of the solution to stationary state
distribution is studied. In particular it is proved the boundedness
respectively the divergence of the fractional order moments of the solution
below respectively above some critical exponent. This exponent is computed. In
particular models it is the heavy tail exponent. When the equation is linear
this exponent determines a new family of weak topologies (stronger compared to
the classical one), related to the convergence to the stationary state.
