  Two types of random evolution processes are studied for ensembles of the
standard map with driving parameter $K$ that determines its degree of
stochasticity. For one type of processes the parameter $K$ is chosen at random
from a Gaussian distribution and is then kept fixed, while for the other type
it varies from step to step. In addition, noise that can be arbitrarily weak is
added. The ensemble average and the average over noise of the diffusion
coefficient is calculated for both types of processes. These two types of
processes are relevant for two types of experimental situations as explained in
the paper. Both types of processes destroy fine details of the dynamics, and
the second process is found to be more effective in destroying the fine
details. We hope that this work is a step in the efforts for developing a
statistical theory for systems with mixed phase space (regular in some parts
and chaotic in other parts).
