The Annals of Probability

Quenched invariance principle for the Knudsen stochastic billiard in a random tube

Francis Comets, Serguei Popov, Gunter M. Schütz, and Marina Vachkovskaia

Full-text: Open access


We consider a stochastic billiard in a random tube which stretches to infinity in the direction of the first coordinate. This random tube is stationary and ergodic, and also it is supposed to be in some sense well behaved. The stochastic billiard can be described as follows: when strictly inside the tube, the particle moves straight with constant speed. Upon hitting the boundary, it is reflected randomly, according to the cosine law: the density of the outgoing direction is proportional to the cosine of the angle between this direction and the normal vector. We also consider the discrete-time random walk formed by the particle’s positions at the moments of hitting the boundary. Under the condition of existence of the second moment of the projected jump length with respect to the stationary measure for the environment seen from the particle, we prove the quenched invariance principles for the projected trajectories of the random walk and the stochastic billiard.

Article information

Ann. Probab. Volume 38, Number 3 (2010), 1019-1061.

First available in Project Euclid: 2 June 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments
Secondary: 37D50: Hyperbolic systems with singularities (billiards, etc.) 60J05: Discrete-time Markov processes on general state spaces 60J25: Continuous-time Markov processes on general state spaces

Cosine law Knudsen random walk stochastic homogenization invariance principle random medium random conductances random walks in random environment


Comets, Francis; Popov, Serguei; Schütz, Gunter M.; Vachkovskaia, Marina. Quenched invariance principle for the Knudsen stochastic billiard in a random tube. Ann. Probab. 38 (2010), no. 3, 1019--1061. doi:10.1214/09-AOP504.

Export citation


  • [1] Barlow, M. T. (2004). Random walks on supercritical percolation clusters. Ann. Probab. 32 3024–3084.
  • [2] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [3] Berger, N. and Biskup, M. (2007). Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Related Fields 137 83–120.
  • [4] Biskup, M. and Prescott, T. M. (2007). Functional CLT for random walk among bounded random conductances. Electron. J. Probab. 12 1323–1348.
  • [5] Bolthausen, E. and Sznitman, A.-S. (2002). Ten Lectures on Random Media. DMV Seminar 32. Birkhäuser, Basel.
  • [6] Comets, F., Popov, S., Schütz, G. M. and Vachkovskaia, M. (2009). Billiards in a general domain with random reflections. Arch. Ration. Mech. Anal. 191 497–537. [Erratum: Arch. Ration. Mech. Anal. 193 (2009) 737–738.]
  • [7] Comets, F., Popov, S., Schütz, G. M. and Vachkovskaia, M. (2009). Transport diffusion coefficient for Knudsen gas in random tube. Preprint. Available at
  • [8] Coppens, M.-C. and Dammers, A. J. (2006). Effects of heterogeneity on diffusion in nanopores. From inorganic materials to protein crystals and ion channels. Fluid Phase Equilibria 241 308–316.
  • [9] Coppens, M.-O. and Malek, K. (2003). Dynamic Monte-Carlo simulations of diffusion limited reactions in rough nanopores. Chem. Eng. Sci. 58 4787–4795.
  • [10] De Masi, A., Ferrari, P. A., Goldstein, S. and Wick, W. D. (1989). An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Stat. Phys. 55 787–855.
  • [11] Durrett, R. (2005). Probability: Theory and Examples, 3rd ed. Duxbury Press, Belmont, CA.
  • [12] Feres, R. and Yablonsky, G. (2004). Knudsen’s cosine law and random billiards. Chem. Eng. Sci. 59 1541–1556.
  • [13] Fontes, L. R. G. and Mathieu, P. (2006). On symmetric random walks with random conductances on ℤd. Probab. Theory Related Fields 134 565–602.
  • [14] Faggionato, A., Schulz-Baldes, H. and Spehner, D. (2006). Mott law as lower bound for a random walk in a random environment. Comm. Math. Phys. 263 21–64.
  • [15] Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 1–19.
  • [16] Komorowski, T., Landim, C. and Olla, S. (2008). Fluctuations in Markov processes. Available at
  • [17] Kozlov, S. M. (1985). The averaging method and walks in inhomogeneous environments. Uspekhi Mat. Nauk 40 61–120, 238.
  • [18] Mathieu, P. and Piatnitski, A. (2007). Quenched invariance principles for random walks on percolation clusters. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 2287–2307.
  • [19] Mathieu, P. (2008). Quenched invariance principles for random walks with random conductances. J. Stat. Phys. 130 1025–1046.
  • [20] Menshikov, M. V., Vachkovskaia, M. and Wade, A. R. (2008). Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains. J. Stat. Phys. 132 1097–1133.
  • [21] Russ, S., Zschiegner, S., Bunde, A. and Kärger, J. (2005). Lambert diffusion in porous media in the Knudsen regime: Equivalence of self- and transport diffusion. Phys. Rev. E 72 030101(R).
  • [22] Sidoravicius, V. and Sznitman, A.-S. (2004). Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Related Fields 129 219–244.
  • [23] Zschiegner, S., Russ, S., Bunde, A., Coppens, M.-O. and Kärger, J. (2007). Normal and anomalous Knudsen diffusion in 2D and 3D channel pores. Diff. Fund. 7 17.