From a kinetic equation to a diffusion under an anomalous scaling
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 4, p. 1301-1322

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}× 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))ds, where |v|1 for small k. We prove that the rescaled process (NlnN) -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.

Une équation de Boltzmann linéaire est interprétée comme équation de Fokker-Planck associée à la densité de probabilité d’un processus de Markov (K(t),i(t),Y(t)) sur (𝕋 2 ×{1,2}× 2 ), où 𝕋 2 est le tore bidimensionnel. Le processus Markovien (K(t),i(t)) est ici un processus de sauts réversible avec des temps d’attente entre deux sauts à moyenne finie mais variance infinie. Y(t) est une fonctionnelle additive de K, définie par Y(t)= 0 t v(K(s))ds, où |v|1 pour k petit. Nous prouvons que le processus (NlnN) -1/2 Y(Nt) converge en distribution vers un mouvement brownien bidimensionnel. En conséquence, et moyennant un changement d’échelle approprié, la solution de l’équation de Boltzmann converge vers celle d’ une équation de diffusion.

DOI : https://doi.org/10.1214/13-AIHP554
Classification:  82C44,  60K35,  60G70
Keywords: anomalous thermal conductivity, kinetic limit, invariance principle
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     author = {Basile, Giada},
     title = {From a kinetic equation to a diffusion under an anomalous scaling},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {4},
     year = {2014},
     pages = {1301-1322},
     doi = {10.1214/13-AIHP554},
     zbl = {06377555},
     mrnumber = {3269995},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_4_1301_0}
}
Basile, Giada. From a kinetic equation to a diffusion under an anomalous scaling. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 4, pp. 1301-1322. doi : 10.1214/13-AIHP554. http://www.numdam.org/item/AIHPB_2014__50_4_1301_0/

[1] O. Aalen. Weak convergence of stochastic integrals related to counting processes. Z. Wahrsch. Verw. Gebiete 38 (1977) 261-277. | MR 448552 | Zbl 0339.60054

[2] K. Aoki, J. Lukkarinen and H. Spohn. Energy transport in weakly anharmonic chain. J. Stat. Phys. 124 (2006) 1105-1129. | MR 2265846 | Zbl 1135.82326

[3] G. Basile and A. Bovier. Convergence of a kinetic equation to a fractional diffusion equation. Markov Process. Related Fields 16 (2010) 15-44. | MR 2664334 | Zbl 1198.82052

[4] G. Basile, C. Bernardin and S. Olla. Thermal conductivity for a momentum conserving model. Comm. Math. Phys. 287 (1) (2009) 67-98. | MR 2480742 | Zbl 1178.82070

[5] G. Basile, S. Olla and H. Spohn. Energy transport in stochastically perturbed lattice dynamics. Arch. Ration. Mech. Anal. 195 (1) (2009) 171-203. | MR 2564472 | Zbl 1187.82017

[6] L. Bertini and B. Zegarlinsky. Coercive inequalities for Kawasaki dynamics. The product case. Markov Process. Related Fields 5 (1999) 125-162. | MR 1762171 | Zbl 0934.60096

[7] P. Billingsley. Convergence of Probability Measures, 2nd edition. Wiley-Interscience, New York, 1999. | MR 1700749 | Zbl 0172.21201

[8] A. Dhar. Heat transport in low-dimensional systems. Adv. Phys. 57 (2008) 457-537. DOI:10.1080/00018730802538522.

[9] R. Durrett and S. Resnick. Functional limit theorems for dependent variables. Ann. Probab. 6 (1978) 829-849. | MR 503954 | Zbl 0398.60024

[10] A. Dvoretzky. Central limit theorems for dependent random variables. In Actes, Congrès int. Math. Tome 2 565-570. Gauthier-Villars, Paris, 1970. | MR 420787 | Zbl 0254.60014

[11] A. Dvoretzky. Asymptotic normality for sums of dependent random variables. In Proc. Sixth Berkeley Symp. Math. Statist. Probability 513-535. Univ. California Press, Berkeley, 1972. | MR 415728 | Zbl 0256.60009

[12] D. Freedman. Brownian Motion and Diffusion. Holden-Day, San Francisco, 1971. | MR 297016 | Zbl 0231.60072

[13] D. Freedman. On tail probabilities for martingales. Ann. Probab. 3 (1975) 100-118. | MR 380971 | Zbl 0313.60037

[14] B. V. Gnedenko and A. N. Kolmogorov. Predel'nye raspredeleniya dlya summ nezavisimyh slučaĭnyh veličin. (Russian) [Limit Distributions for Sums of Independent Random Variables]. Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1949. | MR 41377

[15] M. Jara, T. Komorowski and S. Olla. Limit theorems for additive functionals of a Markov chain. Ann. Appl. Probab. 19 (6) (2009) 2270-2300. | MR 2588245 | Zbl 1232.60018

[16] S. Ghosh, W. Bao, D. L. Nika, S. Subrina, E. P. Pokatilov, C. N. Lau and A. A. Balandin. Dimensional crossover of thermal transport in few-layer graphene materials. Nature Materials 9 (2010) 555-558.

[17] I. S. Helland. Central limit theorems for martingales with discrete or continuous time. Scand. J. Statist. 9 (1982) 79-94. | MR 668684 | Zbl 0486.60023

[18] R. Lefevere and A. Schenkel. Normal heat conductivity in a strongly pinned chain of anharmonic oscillators. J. Stat. Mech. 2006 (2006) L02001. DOI:10.1088/1742-5468/2006/02/L02001.

[19] S. Lepri, R. Livi and A. Politi. Thermal conduction in classical low-dimensional lattice. Phys. Rep. 377 (2003) 1-80. | MR 1978992

[20] T. M. Liggett. L 2 rates of convergence for attractive reversible nearest particle systems. Ann. Probab. 19 (1991) 935-959. | MR 1112402 | Zbl 0737.60092

[21] J. Lukkarinen and H. Spohn. Kinetic limit for wave propagation in a random medium. Arch. Ration. Mech. Anal. 183 (2007) 93-162. | MR 2259341 | Zbl 1176.60053

[22] J. Lukkarinen and H. Spohn. Anomalous energy transport in the FPU-β chain. Comm. Pure Appl. Math. 61 (2008) 1753-1789. | MR 2456185 | Zbl 1214.82057

[23] D. L. Mcleish. Dependent central limit theorems and invariance principles. Ann. Probab. 2 (4) (1974) 620-628. | MR 358933 | Zbl 0287.60025

[24] A. Mellet, S. Mischler and C. Mouhot. Fractional diffusion limit for collitional kinetic equations. Arch. Ration. Mech. Anal. 199 (2) (2011) 493-525. | MR 2763032 | Zbl 1294.82033

[25] S. P. Meyn and R. L. Tweedie. Markov Chains and Stochastic Stability, 2nd edition. Cambridge Univ. Press, Cambridge, 2009. | MR 2509253 | Zbl 1165.60001

[26] R. E. Peierls. Zur kinetischen Theorie der Waermeleitung in Kristallen. Ann. Phys. 3 (1929) 1055-1101. | JFM 55.0547.01

[27] A. Pereverzev. Fermi-Pasta-Ulam β lattice: Peierls equation and anomalous heat conductivity. Phys. Rev. E 68 (2003) 056124. | MR 2060102

[28] M. Röckner and F.-Y. Wang. Weak Poincaré inequalities and L 2 -convergence rates of Markov semigroups. J. Funct. Anal. 185 (2001) 564-603. | MR 1856277 | Zbl 1009.47028

[29] S. J. Sepanski. Some invariance principles for random vectors in the generalized domain of attraction of the multivariate normal law. J. Theoret. Probab. 10 (4) (1997) 153-1063. | MR 1481659 | Zbl 0897.60039

[30] H. Spohn. The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics. J. Stat. Phys. 124 (2-4) (2006) 1041-1104. | MR 2264633 | Zbl 1106.82033