**J. A. Carrillo** and** G. Toscani**

*Contractive probability metrics and asymptotic behavior of dissipative kinetic equations*

**Pages** 75-198

**Received:** 8 January 2007

**Mathematics Subject Classification (2000):** 82C40 - 35B40

**Abstract**
The present notes are intended to present a detailed review of the
existing results in dissipative kinetic theory which make use of
the contraction properties of two main families of probability
metrics: optimal mass transport and Fourier-based metrics. The
first part of the notes is devoted to a self-consistent summary
and presentation of the properties of both probability metrics,
including new aspects on the relationships between them and other
metrics of wide use in probability theory. These results are of
independent interest with potential use in other contexts in
Partial Differential Equations and Probability Theory. The second
part of the notes makes a different presentation of the asymptotic
behavior of Inelastic Maxwell Models than the one presented in the
literature and it shows a new example of application: particle's
bath heating. We show how starting from the contraction properties
in probability metrics, one can deduce the existence, uniqueness
and asymptotic stability in classical spaces. A global strategy
with this aim is set up and applied in two dissipative models.

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