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D. Matthes:
"Kinetic models with random collision kernels on the line";
Talk: Seminar of the Department of Statistics, Univ. Toulouse, Université Paul Sabatier, Toulouse. (invited); 2008-12-09.

Kinetic equations model the redistribution of energy and momentum in an ensemble of particles due to binary collisions. The most famous example is the Boltzmann equation, where the collisions are taken literally, i.e., particles interact according to the laws of classical mechanics. More recently, various alternative interpretations of the collisions became popular; for example, in econophysics, particles are thought of as interacting agents in a closed economy, the particle energy is identified with an agent's wealth, and the binary "collisions" are trades, in which wealth is exchanged according to certain laws, which obviously differ from those of classical mechanics. In particular, some randomness is involved, which corresponds to gains or losses due to risky investments.

In this talk, we will introduce a quite general class of one-dimensional kinetic equations with random collision operators. This class contains both the famous Kac caricature of the Boltzmann equation as well as various models from econophysics. By application of the central limit theorem for triangular arrays, we are able to prove equilibration of the solutions in the long-time regime, and to identify the steady state as a mixture of stable laws. Crucial features of the steady states like the fatness of the over-populated tail are accessible by moment methods. Under natural assumptions, we are further able to show time-uniform propagation of regularity, and thus improve the equilibration from weak to strong convergence.

This is joint work with Giuseppe Toscani and Federico Bassetti from Pavia, as well as Lucia Ladelli from Milano.

Siehe englischen Abstract.

Kinetic equations, equilibration, propagation of smoothness