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D. Matthes:
"Equilibration and Propagation of Smoothness in a Kinetic Model that is Conservative in the Mean";
Keynote Lecture: Workshop on Kinetic Equations: Direct and Inverse Problems, Mantova; 2008-05-14 - 2008-05-17.

A Boltzmann equation on the real line is considered. The collision
kernel is stochastic and conserves mass but not any higher momentum
in individual collisions. However, the first momentum is conserved
in the statistical mean. The equation possesses an interpretation
in terms of trade interactions between agents on a risky market.

Using Fourier metrics, we prove convergence of weak solutions to a
steady state under mild hypotheses on the random kernel and initial
condition. The steady state's tail shape (fat or exponentially small)
is classified by inspection of the hierarchy of moment equations.

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 englisches Abstract.

Boltzmann equation, Econophysics