Convergence to Equilibrium of the Solution of Kac's Kinetic Equation. A Probabilistic View

Eugenio Regazzini

Bollettino dell'Unione Matematica Italiana (2009)

  • Volume: 2, Issue: 1, page 175-198
  • ISSN: 0392-4041

Abstract

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Let f ⁢ ( ⋅ , t ) be the probability density function representing the solution of Kac's Boltzmann-like equation at time t , with initial data f 0 , and let g σ be the Gaussian density with zero mean and variance σ 2 , σ 2 being the value of the second moment of f 0 . Henry McKean Jr. put forward the conjecture that the total variation distance between f ⁢ ( ⋅ , t ) and g σ goes to zero, as t → + ∞ , with an exponential rate equal to - 1 / 4 . This lecture aims at explaining the main efforts made to a view to validating this conjecture, and concludes with the theorem stating that this holds true whenever f 0 has finite fourth moment and its Fourier transform φ 0 satisfies | φ 0 ⁢ ( ξ ) | = o ⁢ ( | ξ | - p ) as | ξ | → + ∞ , for some p > 0 . The first part of the lecture expounds the derivation of the Kac Boltzmann-like equation from the Kac master equation. A detailed description of the probabilistic methods resorted to prove the above-mentioned theorem is then given. The final part mentions further applications of these methods to other kinetic models.

How to cite

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Regazzini, Eugenio. "Convergence to Equilibrium of the Solution of Kac's Kinetic Equation. A Probabilistic View." Bollettino dell'Unione Matematica Italiana 2.1 (2009): 175-198. <http://eudml.org/doc/290569>.

@article{Regazzini2009,
abstract = {Let $f(\cdot, t)$ be the probability density function representing the solution of Kac's Boltzmann-like equation at time $t$, with initial data $f_\{0\}$, and let $g_\{\sigma\}$ be the Gaussian density with zero mean and variance $\sigma^\{2\}$, $\sigma^\{2\}$ being the value of the second moment of $f_\{0\}$. Henry McKean Jr. put forward the conjecture that the total variation distance between $f(\cdot,t)$ and $g_\{\sigma\}$ goes to zero, as $t \to + \infty$, with an exponential rate equal to $-1/4$. This lecture aims at explaining the main efforts made to a view to validating this conjecture, and concludes with the theorem stating that this holds true whenever $f_\{0\}$ has finite fourth moment and its Fourier transform $\varphi_\{0\}$ satisfies $|\varphi_\{0\}(\xi)| = o(|\xi|^\{-p\})$ as $|\xi| \to + \infty$, for some $p > 0$. The first part of the lecture expounds the derivation of the Kac Boltzmann-like equation from the Kac master equation. A detailed description of the probabilistic methods resorted to prove the above-mentioned theorem is then given. The final part mentions further applications of these methods to other kinetic models.},
author = {Regazzini, Eugenio},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {175-198},
publisher = {Unione Matematica Italiana},
title = {Convergence to Equilibrium of the Solution of Kac's Kinetic Equation. A Probabilistic View},
url = {http://eudml.org/doc/290569},
volume = {2},
year = {2009},
}

TY - JOUR
AU - Regazzini, Eugenio
TI - Convergence to Equilibrium of the Solution of Kac's Kinetic Equation. A Probabilistic View
JO - Bollettino dell'Unione Matematica Italiana
DA - 2009/2//
PB - Unione Matematica Italiana
VL - 2
IS - 1
SP - 175
EP - 198
AB - Let $f(\cdot, t)$ be the probability density function representing the solution of Kac's Boltzmann-like equation at time $t$, with initial data $f_{0}$, and let $g_{\sigma}$ be the Gaussian density with zero mean and variance $\sigma^{2}$, $\sigma^{2}$ being the value of the second moment of $f_{0}$. Henry McKean Jr. put forward the conjecture that the total variation distance between $f(\cdot,t)$ and $g_{\sigma}$ goes to zero, as $t \to + \infty$, with an exponential rate equal to $-1/4$. This lecture aims at explaining the main efforts made to a view to validating this conjecture, and concludes with the theorem stating that this holds true whenever $f_{0}$ has finite fourth moment and its Fourier transform $\varphi_{0}$ satisfies $|\varphi_{0}(\xi)| = o(|\xi|^{-p})$ as $|\xi| \to + \infty$, for some $p > 0$. The first part of the lecture expounds the derivation of the Kac Boltzmann-like equation from the Kac master equation. A detailed description of the probabilistic methods resorted to prove the above-mentioned theorem is then given. The final part mentions further applications of these methods to other kinetic models.
LA - eng
UR - http://eudml.org/doc/290569
ER -

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