Characterization of weak convergence of probability-valued solutions of general one-dimensional kinetic equations
arXiv:1404.6124 · doi:10.1007/s10955-015-1200-6
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.
References in corpus (9)
- Pareto Law in a Kinetic Model of Market with Random Saving Propensity
- Reaching the best possible rate of convergence to equilibrium for solutions of Kac's equation via central limit theorem
- Probabilistic study of the speed of approach to equilibrium for an inelastic Kac model
- Self-similar solutions in one-dimensional kinetic models: A probabilistic view
- The role of the central limit theorem in discovering sharp rates of convergence to equilibrium for the solution of the Kac equation
- Central limit theorem for the solution of the Kac equation
- Complete characterization of convergence to equilibrium for an inelastic Kac model
- Probabilistic View of Explosion in an Inelastic Kac Model
- Infinite energy solutions to inelastic homogeneous Boltzmann equation