Weak convergence of probability-valued solutions of general one-dimensional kinetic equations: Characterizations and Applications

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. An entire section is devoted to an application of these results to the distribution of income. It highlights a strict relationship between the Arrow-de Finetti local risk aversion index, assumed to be the same for all agents, and the inequality (concentration) in the stationary income distribution.