  We are interested in stationary "fluid" random evolutions with independent
increments. Under some mild assumptions, we show they are solutions of a
stochastic differential equation (SDE). There are situations where these
evolutions are not described by flows of diffeomorphisms, but by coalescing
flows or by flows of probability kernels. In an intermediate phase, for which
there exist a coalescing flow and a flow of kernels solution of the SDE, a
classification is given: All solutions of the SDE can be obtained by filtering
a coalescing motion with respect to a subnoise containing the Gaussian part of
its noise. Thus, the coalescing motion cannot be described by a white noise.
