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Clustering statistics for sticky particles with Brownian initial velocity. (English) Zbl 0959.60074

Consider a linear model of clustering of particles which evolves according to the Burgers equation: \(\partial_tu+ u\partial_xu= 0\), \(u(t,x)\) denoting the velocity at time \(t\) of the particle located at \(x\). This describes a model where (clusters of) particles which shock together aggregate, with conservation of mass and momentum. Start at \(t=0\) with uniform distribution of mass (on \(\mathbb{R}\)) and with Brownian distribution of velocities \(u(x,0)\) (on \(\mathbb{R}_+\)). The main result asserts that, conditionally on the state of the system at time \(t>0\), the masses of the clusters at time \(s< t\) are obtained by breaking into peaces each cluster of the time \(t\), independently of all locations and velocities and of the masses of the other clusters, the law of these peaces being governed by some precise Poisson measure on \(\mathbb{R}^*_+\). Moreover, as \(s\) decreases and \(t\) stays fixed, the above description of clusters evolves as a sequence of independent Brownian fragmentation processes, which is precisely related to the so-called standard additive coalescent.

MSC:

60J65 Brownian motion
60J25 Continuous-time Markov processes on general state spaces
35Q53 KdV equations (Korteweg-de Vries equations)
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