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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.Comment: Published at http://dx.doi.org/10.1214/009117904000000207 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The purpose of this article is to introduce and study a relativistic motion whose acceleration, in proper time, is given by a white noise. We deal with general relativity, and consider more closely the problem of the asymptotic behaviour of paths in the Schwarzschild geometry example. 1 Corpora cum deorsum rectum per inane feruntur, ponderibus propriis incerto tempore ferme incertisque locis spatio depellere paulum, tantum quod momen mutatum dicere possis. Quod nisi declinare solerent, omnia deorsum, imbris uti guttae, caderent per inane profundum, nec foret offensus natus, nec plaga creata principiis : ita nil umquam natura creasset. Lucrecius

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