  We introduce and discuss spatially homogeneous Maxwell-type models of the
nonlinear Boltzmann equation undergoing binary collisions with a random
component. The random contribution to collisions is such that the usual
collisional invariants of mass, momentum and energy do not hold pointwise, even
if they all hold in the mean. Under this assumption it is shown that, while the
Boltzmann equation has the usual conserved quantities, it possesses a steady
state with power-like tails for certain random variables. A similar situation
occurs in kinetic models of economy recently considered by two of the authors
[24], which are conservative in the mean but possess a steady distribution with
Pareto tails. The convolution-like gain operator is subsequently shown to have
good contraction/expansion properties with respect to different metrics in the
set of probability measures. Existence and regularity of isotropic stationary
states is shown directly by constructing converging iteration sequences as done
in [8]. Uniqueness, asymptotic stability and estimates of overpopulated high
energy tails of the steady profile are derived from the basic property of
contraction/expansion of metrics. For general initial conditions the solutions
of the Boltzmann equation are then proved to converge with computable rate as t
goes to infinity to the steady solution in these distances, which metricizes
the weak convergence of measures. These results show that power-like tails in
Maxwell models are obtained when the point-wise conservation of momentum and/or
energy holds only globally.
