Abstract:
Originally, kinetic equations have been devised to model the
redistribution of energy and momentum in an ensemble of
particles which interact in binary collisions. Recently, some
alternative interpretations (instead of particles exchanging
energy, one considers e.g. financial traders exchanging
money) have produced kinetic equations with an additional
stochastic mechanism for gains and losses in the energy in
particle interactions. The behavior of solutions is quite different
from that of the physical original. Most notably, the stationary
velocity distribution, which is always Gaussian in the physical
setting, may now possess an algebraically fat tail.

In this talk, we study this novel class of kinetic equations in
view of their equilibration behavior and the shape of the steady
state. A probabilistic representation of the solution is derived,
and weak long-time convergence is obtained by application of
the central limit theorem. Weak convergence can be upgraded
to strong L1-convergence under additional conditions. The
speed of convergence can be estimated in Wasserstein metrics.

This is joint work with Federico Bassetti (UNIPV), Lucia Ladelli
(POLIMI) and Giuseppe Toscani (UNIPV).