Coalescing and noncoalescing stochastic flows in $$R_ 1$$.(English)Zbl 0536.60016

Two kinds of Brownian stochastic flows on the real line are identified and studied. In one kind the constituent random mappings are all homeomorphisms. The other uses random mappings whose ranges are locally finite points under the flow then ”coalesce”. Conditions are given which imply one or the other kind of behaviour; these conditions depend on the differentiability at zero of a covariance function used to characterise the flow. Whether other kinds of behaviour are possible is left as an open question, as is the matter of whether coalescence can occur from Brownian flows in the plane or space. Various properties are investigated, and the case of exponential covariance is shown to lead to a spatial Markov property.
Reviewer: Wilfrid S.Kendall

MSC:

 60B99 Probability theory on algebraic and topological structures 60J60 Diffusion processes 60G99 Stochastic processes
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References:

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