The Annals of Applied Probability

Reaching the best possible rate of convergence to equilibrium for solutions of Kac’s equation via central limit theorem

Emanuele Dolera, Ester Gabetta, and Eugenio Regazzini

Full-text: Open access


Let f(⋅, t) be the probability density function which represents the solution of Kac’s equation at time t, with initial data f0, and let gσ be the Gaussian density with zero mean and variance σ2, σ2 being the value of the second moment of f0. This is the first study which proves that the total variation distance between f(⋅, t) and gσ goes to zero, as t→+∞, with an exponential rate equal to −1/4. In the present paper, this fact is proved on the sole assumption that f0 has finite fourth moment and its Fourier transform ϕ0 satisfies |ϕ0(ξ)|=o(|ξ|p) as |ξ|→+∞, for some p>0. These hypotheses are definitely weaker than those considered so far in the state-of-the-art literature, which in any case, obtains less precise rates.

Article information

Ann. Appl. Probab., Volume 19, Number 1 (2009), 186-209.

First available in Project Euclid: 20 February 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 82C40: Kinetic theory of gases

Berry–Esseen inequalities central limit theorem Kac’s equation total variation distance Wild’s sum


Dolera, Emanuele; Gabetta, Ester; Regazzini, Eugenio. Reaching the best possible rate of convergence to equilibrium for solutions of Kac’s equation via central limit theorem. Ann. Appl. Probab. 19 (2009), no. 1, 186--209. doi:10.1214/08-AAP538.

Export citation


  • Apostol, T. M. (1974). Mathematical Analysis, 2nd ed. Addison-Wesley, Reading, MA.
  • Bassetti, F., Gabetta, E. and Regazzini, E. (2007). On the depth of the trees in the McKean representation of Wild’s sums. Transport Theory Statist. Phys. 36 421–438.
  • Beurling, A. (1939). Sur les intégrales de Fourier absolument convergentes et leur application à une trasformation fonctionelle. In 9th Congr. Math. Scandinaves (Helsinki, 1938) 199–210. Tryekeri, Helsinki. See also The Collected Works of Arne Beurling. Vol. 2. Harmonic Analysis (L. Carleson, P. Malliavin, V. Neuberger and J. Wermer, eds.). Birkhäuser, Boston, 1989.
  • Carlen, E. A., Carvalho, M. C. and Gabetta, E. (2000). Central limit theorem for Maxwellian molecules and truncation of the Wild expansion. Comm. Pure Appl. Math. 53 370–397.
  • Carlen, E. A., Carvalho, M. C. and Gabetta, E. (2005). On the relation between rates of relaxation and convergence of Wild sums for solutions of the Kac equation. J. Funct. Anal. 220 362–387.
  • Carlen, E., Gabetta, E. and Regazzini, E. (2008). Probabilistic investigations on the explosion of solutions of the Kac equation with infinite energy initial distribution. J. Appl. Probab. 45 95–106.
  • Carlen, E. A., Gabetta, E. and Toscani, G. (1999). Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas. Comm. Math. Phys. 199 521–546.
  • Carlen, E. A. and Lu, X. (2003). Fast and slow convergence to equilibrium for Maxwellian molecules via Wild sums. J. Statist. Phys. 112 59–134.
  • Cercignani, C. (1975). Theory and Application of the Boltzmann Equation. Elsevier, New York.
  • Chow, Y. S. and Teicher, H. (1997). Probability Theory, 3rd ed. Springer, New York.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications. I, 3rd ed. Wiley, New York.
  • Folland, G. B. (1999). Real Analysis, 2nd ed. Wiley, New York.
  • Gabetta, E. and Regazzini, E. (2006a). Some new results for McKean’s graphs with applications to Kac’s equation. J. Statist. Phys. 125 947–974.
  • Gabetta, E. and Regazzini, E. (2006b). Central limit theorem for the solution of the Kac equation: Speed of approach to equilibrium in weak metrics. Preprint IMATI-CNR 27-PV, Pavia. Probab. Theory Related Fields. To appear.
  • Gabetta, E. and Regazzini, E. (2008). Central limit theorem for the solution of the Kac equation. Preprint IMATI-CNR 26-PV, Pavia. Ann. Appl. Probab. 18 2320–2336.
  • Kac, M. (1956). Foundations of kinetic theory. In Proc. Third Berkeley Symp. Math. Statist. Probab. 1954–1955 3 171–197. Univ. California Press, Berkeley.
  • Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • Linnik, J. V. (1959). An information-theoretic proof of the central limit theorem with Lindeberg conditions. Theory Probab. Appl. 4 288–299.
  • Matthes, D. and Toscani, G. (2008). On steady distributions of kinetic models of conservative economies. J. Statist. Phys. 130 1087–1117.
  • McKean, H. P. Jr. (1966). Speed of approach to equilibrium for Kac’s caricature of a Maxwellian gas. Arch. Rational Mech. Anal. 21 343–367.
  • McKean, H. P. Jr. (1967). An exponential formula for solving Boltmann’s equation for a Maxwellian gas. J. Combinatorial Theory 2 358–382.
  • Merris, R. (2003). Combinatorics, 2nd ed. Wiley, New York.
  • Morgenstern, D. (1954). General existence and uniqueness proof for spatially homogeneous solutions of the Maxwell–Boltzmann equation in the case of Maxwellian molecules. Proc. Natl. Acad. Sci. USA 40 719–721.
  • Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, New York.
  • Saulis, L. and Statulevičius, V. A. (1991). Limit Theorems for Large Deviations. Kluwer, Dordrecht.
  • Truesdell, C. and Muncaster, R. G. (1980). Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas. Pure and Applied Mathematics 83. Academic Press, New York.
  • Wild, E. (1951). On Boltzmann’s equation in the kinetic theory of gases. Proc. Cambridge Philos. Soc. 47 602–609.
  • Villani, C. (2006). Mathematics of granular materials. J. Statist. Phys. 124 781–822.
  • Villani, C. (2008). Entropy production and convergence to equilibrium. In Entropy Methods for the Boltzmann Equation. Lecture Notes in Mathematics 1916 1–70. Springer, Berlin.
  • Zolotarev, V. M. (1997). Modern Theory of Summation of Random Variables. VSP, Utrecht.