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D. Matthes:
"Kinetic models for wealth redistribution. Equilibration and propagation of smoothness";
Keynote Lecture: Workshop "Kinetic modelling for socio-economic and related problems", Vigevano, Italien (invited); 2008-11-27 - 2008-11-29.

A class of kinetic models for the redistribution of wealth in a simple market economy is discussed. Molecules and their velocities are interpreted as agents and their wealth; the collisional operator then describes how wealth changes hands in binary trade interactions. The two crucial ingredients for the model are a saving propensity and randomness. While the first corresponds to the agents´ tendency to retain a certain amount of their wealth in trades, the second refers to risky investments.

In contrast to the classical Boltzmann equation, energy is generally not conserved in collisions, and the linear momentum (or rather, the mean wealth) is not strictly conserved but only in the statistical mean. This freedom in the modelling leads to a rich variety of possible steady states for the kinetic equation. In particular, we are able to identify models which give rise to fat tailed stationary wealth distribution, corresponding to the formation of a rich high society.

Using Fourier metrics, we prove convergence of weak solutions to a steady state under mild hypotheses on the random kernel and initial condition. Moreover, propagation of Sobolev regularity, and Gevrey smoothness of the steady state is shown by combining the classical result for Maxwell molecules (Carlen/Gabetta/Toscani´99) with methods recently developed for the Kac equation (Desvillettes/Furioli/Terraneo´06).

Siehe englischen Abstract.

kinetic equations, wealth distribution, Pareto tails