From a kinetic equation to a diffusion under an anomalous scaling

Giada Basile

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 4, page 1301-1322
  • ISSN: 0246-0203

Abstract

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A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process ( K ( t ) , i ( t ) , Y ( t ) ) on ( 𝕋 2 × { 1 , 2 } × ℝ 2 ) , where 𝕋 2 is the two-dimensional torus. Here ( K ( t ) , i ( t ) ) is an autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance. Y ( t ) is an additive functional of K , defined as ∫ 0 t v ( K ( s ) ) d s , where | v | ∼ 1 for small k . We prove that the rescaled process ( N ln N ) - 1 / 2 Y ( N t ) converges in distribution to a two-dimensional Brownian motion. As a consequence, the appropriately rescaled solution of the Boltzmann equation converges to the solution of a diffusion equation.

How to cite

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Basile, Giada. "From a kinetic equation to a diffusion under an anomalous scaling." Annales de l'I.H.P. Probabilités et statistiques 50.4 (2014): 1301-1322. <http://eudml.org/doc/272059>.

@article{Basile2014,
abstract = {A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process $(K(t),i(t),Y(t))$ on $(\mathbb \{T\} ^\{2\}\times \lbrace 1,2\rbrace \times \mathbb \{R\} ^\{2\})$, where $\mathbb \{T\} ^\{2\}$ is the two-dimensional torus. Here $(K(t),i(t))$ is an autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance. $Y(t)$ is an additive functional of $K$, defined as $\int _\{0\}^\{t\}v(K(s))\,\mathrm \{d\}s$, where $|v|\sim 1$ for small $k$. We prove that the rescaled process $(N\ln N)^\{-1/2\}Y(Nt)$ converges in distribution to a two-dimensional Brownian motion. As a consequence, the appropriately rescaled solution of the Boltzmann equation converges to the solution of a diffusion equation.},
author = {Basile, Giada},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {anomalous thermal conductivity; kinetic limit; invariance principle},
language = {eng},
number = {4},
pages = {1301-1322},
publisher = {Gauthier-Villars},
title = {From a kinetic equation to a diffusion under an anomalous scaling},
url = {http://eudml.org/doc/272059},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Basile, Giada
TI - From a kinetic equation to a diffusion under an anomalous scaling
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 4
SP - 1301
EP - 1322
AB - A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process $(K(t),i(t),Y(t))$ on $(\mathbb {T} ^{2}\times \lbrace 1,2\rbrace \times \mathbb {R} ^{2})$, where $\mathbb {T} ^{2}$ is the two-dimensional torus. Here $(K(t),i(t))$ is an autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance. $Y(t)$ is an additive functional of $K$, defined as $\int _{0}^{t}v(K(s))\,\mathrm {d}s$, where $|v|\sim 1$ for small $k$. We prove that the rescaled process $(N\ln N)^{-1/2}Y(Nt)$ converges in distribution to a two-dimensional Brownian motion. As a consequence, the appropriately rescaled solution of the Boltzmann equation converges to the solution of a diffusion equation.
LA - eng
KW - anomalous thermal conductivity; kinetic limit; invariance principle
UR - http://eudml.org/doc/272059
ER -

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