Characterization of Weak Convergence of Probability-Valued Solutions of General One-Dimensional Kinetic Equations

Abstract

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. Primary 60F05, 82C40; secondary 91B15.

Inequality and risk aversion in economies open to altruistic attitudes

“…Remark 3. As far as the extra conditions ( 6) and ( 7) are concerned, they play a mere technical role, that can be noted in the proofs of Theorems 1, 2, 4 and 5 in Perversi and Regazzini (2015). In any case, the meaning of ( 6) is clear, while ( 7) is automatically satisfied whenever the increment ∆ (and not only E(∆)) of the joint welfare function is equal to zero in every contract.…”

“…This section is devoted to the complete formulation of the main results already expounded in Subsection 1.2. In point of fact, our role is limited to rearranging, in a form suitable for the present setting, a few well-known propositions proved in Bassetti et al (2011), Bassetti and Perversi (2013) and Perversi and Regazzini (2015). It is worth recalling that the limit µ ∞ of the solution µ t of (3), as t → +∞, is meant in the sense of weak convergence of p.m.'s: µ t converges weakly to the p.m.…”

“…Coming back to the framework stated in Subsection 1.1.3, one is faced with an intermediate situation, that is, convergence of µ t occurs only in the presence of a distinguished class of initial data µ 0 − essentially, the class of the weak Pareto laws − and the steady states are completely determined in the form of their Fourier-Stieltjes transforms. These statements, derived from Theorems 1 and 4 in Perversi and Regazzini (2015), are condensed into the following Fact 3.2. Assume (H 1 )-(H 2 ) are in force together with the extra conditions that the marginal p.d.…”

“…This equation has been studied extensively in Bassetti et al (2011) followed by Bassetti and Ladelli (2012), Bassetti and Perversi (2013) and Perversi and Regazzini (2015), especially w.r.t. the asymptotic behaviour of µ t , as t goes to infinity.…”

Mean field dynamics of interaction processes with duplication, loss and copy

“…and it is a special case of a general class of kinetic equations which have been deeply investigated, see for instance in [2,3,14,40,44]. It is immediate to realize that the interactions (44) correspond to a mean version of the interactions (1), and can be formally obtained from (1)…”