Coalescing and noncoalescing stochastic flows in R1

We study homogeneous stochastic flows, families Xst, 0 [less-than-or-equals, slant] s [less-than-or-equals, slant] t < [infinity] of random mappings of R1 into itself, with the composition property Xtu o Xst = Xsu, s [less-than-or-equals, slant] t [less-than-or-equals, slant] u, and with independent 'increments'. Depending on the differentiability at 0 of the covariance function of the field of small displacements, the mapping Xst, s and t fixed, may be smooth or may be a step function mapping R1 into a countable set of points with no limit points. (The latter kind of situation has occurred in related work of R. Arratia.) It is not known whether other behavior is possible. Some results in the countable-range case are deduced from duality results obtained for the smooth case. The case of double exponential correlation leads to a moving point process with certain spatial Markovian properties.