Central limit theorems for solutions of the Kac equation: speed of approach to equilibrium in weak metrics

DOI10.1007/s00440-008-0196-0zbMath1181.60030OpenAlexW1974235335MaRDI QIDQ846743

Ester Gabetta, Eugenio Regazzini

Publication date: 9 February 2010

Published in: Probability Theory and Related Fields (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s00440-008-0196-0

zbMATH Keywords

central limit theoremBerry-Esseen inequality\(\chi\)-weighted metricsBoltzmann (Kac) equationGini's (Tanaka, Wasserstein) metricsKolmogorov's metric

Mathematics Subject Classification ID

Central limit and other weak theorems (60F05) Kinetic theory of gases in time-dependent statistical mechanics (82C40)

Related Items (10)

Multi-dimensional smoothing transformations: existence, regularity and stability of fixed points ⋮ Complete characterization of convergence to equilibrium for an inelastic Kac model ⋮ Estimates of the approximation of weighted sums of conditionally independent random variables by the normal law ⋮ Proof of a McKean conjecture on the rate of convergence of Boltzmann-equation solutions ⋮ Probabilistic view of explosion in an inelastic Kac model ⋮ 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 a class of one-dimensional kinetic equations ⋮ 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 ⋮ Characterization of weak convergence of probability-valued solutions of general one-dimensional kinetic equations

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas
• Some new results for McKean's graphs with applications to Kac's equation
• The role of the central limit theorem in discovering sharp rates of convergence to equilibrium for the solution of the Kac equation
• Reaching the best possible rate of convergence to equilibrium for solutions of Kac's equation via central limit theorem
• Exact solutions of the nonlinear Boltzmann equation and the theory of relaxation of a Maxwellian gas
• Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas
• Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation.
• On the relation between rates of relaxation and convergence of wild sums for solutions of the Kac equation
• Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials
• Central limit theorem for the solution of the Kac equation
• On mass transportation
• On a problem of Monge
• Contractive metrics for a Boltzmann equation for granular gases: Diffusive equilibria
• The inverse of the error function
• The tanaka functional and exponential convergence for non cut-off molecules
• An inequality for a functional of probability distributions and its application to Kac's one-dimensional model of a Maxwellian gas
• Inequalities for E k(X, Y) when the marginals are fixed
• Rational Chebyshev Approximations for the Inverse of the Error Function
• Probabilistic treatment of the Boltzmann equation of Maxwellian molecules
• Modern Theory of Summation of Random Variables
• Central limit theorem for Maxwellian molecules and truncation of the wild expansion
• Probabilistic Investigations on the Explosion of Solutions of the KAC Equation with Infinite Energy Initial Distribution
• An exponential formula for solving Boltzmann's equation for a Maxwellian gas
• On the Berry-Esseen theorem
• A non-uniform Berry-Esseen bound via Stein's method

This page was built for publication: Central limit theorems for solutions of the Kac equation: speed of approach to equilibrium in weak metrics