8 research outputs found
Characterization of weak convergence of probability-valued solutions of general one-dimensional kinetic equations
For a general inelastic Kac-like equation recently proposed, this paper
studies the long-time behaviour of its probability-valued solution. In
particular, the paper provides necessary and sufficient conditions for the
initial datum in order that the corresponding solution converges to
equilibrium. The proofs rest on the general CLT for independent summands
applied to a suitable Skorokhod representation of the original solution
evaluated at an increasing and divergent sequence of times. It turns out that,
roughly speaking, the initial datum must belong to the standard domain of
attraction of a stable law, while the equilibrium is presentable as a mixture
of stable laws
Inequality and risk aversion in economies open to altruistic attitudes
This paper attempts to find a relationship between agents' risk aversion and
inequality of incomes. Specifically, a model is proposed for the evolution in
time of surplus/deficit distribution, and the long-time distributions are
characterized almost completely. They turn out to be weak Pareto laws with
exponent linked to the relative risk aversion index which, in turn, is supposed
to be the same for every agent. On the one hand, the aforesaid link is
expressed by an affine transformation. On the other hand, the level of the
relative risk aversion index results from a frequency distribution of
observable quantities stemming from how agents interact in an economic sense.
Combination of these facts is conducive to the specification of qualitative and
quantitative characteristics of actions fit for the control of income
concentration
Probabilistic View of Explosion in an Inelastic Kac Model
Let be the family of probability measures
corresponding to the solution of the inelastic Kac model introduced in
Pulvirenti and Toscani [\textit{J. Stat. Phys.} \textbf{114} (2004) 1453-1480].
It has been proved by Gabetta and Regazzini [\textit{J. Statist. Phys.}
\textbf{147} (2012) 1007-1019] that the solution converges weakly to
equilibrium if and only if a suitable symmetrized form of the initial data
belongs to the standard domain of attraction of a specific stable law. In the
present paper it is shown that, for initial data which are heavier-tailed than
the aforementioned ones, the limiting distribution is improper in the sense
that it has probability 1/2 "adherent" to and probability 1/2
"adherent" to . It is explained in which sense this phenomenon is
amenable to a sort of explosion, and the main result consists in an explicit
expression of the rate of such an explosion. The presentation of these
statements is preceded by a discussion about the necessity of the assumption
under which their validity is proved. This gives the chance to make an
adjustment to a portion of a proof contained in the above-mentioned paper by
Gabetta and Regazzini
Speed of convergence to equilibrium in Wasserstein metrics for Kac-like kinetic equations
This work deals with a class of one-dimensional measure-valued kinetic equations, which constitute extensions of the Kac caricature. It is known that if the initial datum belongs to the domain of normal attraction of an α-stable law, the solution of the equation converges weakly to a suitable scale mixture of centered α-stable laws. In this paper we present explicit exponential rates for the convergence to equilibrium in Kantorovich-Wasserstein distances of order p > α, under the natural assumption that the distance between the initial datum and the limit distribution is finite. For α = 2 this assumption reduces to the finiteness of the absolute moment of order p of the initial datum. On the contrary, when α α. For this case, we provide sufficient conditions for the finiteness of the Kantorovich-Wasserstein distance