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Deterministic approximations of random reflectors

Authors: Omer Angel, Krzysztof Burdzy and Scott Sheffield
Journal: Trans. Amer. Math. Soc. 365 (2013), 6367-6383
MSC (2010): Primary 37D50
Published electronically: June 3, 2013
MathSciNet review: 3105755
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Abstract: Within classical optics, one may add microscopic ``roughness'' to a macroscopically flat mirror so that parallel rays of a given angle are reflected at different outgoing angles. Taking the limit (as the roughness becomes increasingly microscopic) one obtains a flat surface that reflects randomly, i.e., the transition from incoming to outgoing ray is described by a probability kernel (whose form depends on the nature of the microscopic roughness).

We consider two-dimensional optics (a.k.a. billiards) and show that every random reflector on a line that satisfies a necessary measure-preservation condition (well established in the theory of billiards) can be approximated by deterministic reflectors in this way.

References [Enhancements On Off] (What's this?)

  • 1. K. Burdzy, R. Bass, Z. Chen and M. Hairer, Stationary distributions for diffusions with inert drift. Probab. Theory Rel. Fields 146 (2010) 1-47. MR 2550357 (2011b:60226)
  • 2. S. Chandrasekhar, Radiative transfer. Dover Publications, Inc., New York, 1960. MR 0111583 (22:2446)
  • 3. N. Chernov and R. Markarian, Chaotic billiards. Mathematical Surveys and Monographs, 127. American Mathematical Society, Providence, RI, 2006. MR 2229799 (2007f:37050)
  • 4. F. Comets and S. Popov, Ballistic regime for random walks in random environment with unbounded jumps and Knudsen billiards. Ann. Inst. H. Poincaré Probab. Statist. (2012) (to appear).
  • 5. F. Comets, S. Popov, G. Schütz and M. Vachkovskaia, Billiards in a general domain with random reflections. Arch. Rat. Mech. Anal. 191, (2009) 497-537. MR 2481068 (2010h:37077)
  • 6. F. Comets, S. Popov, G. Schütz and M. Vachkovskaia, Erratum: Billiards in a general domain with random reflections. Arch. Ration. Mech. Anal. 193, (2009), 737-738. MR 2525117 (2010h:37078)
  • 7. F. Comets, S. Popov, G. Schütz and M. Vachkovskaia, Quenched invariance principle for the Knudsen stochastic billiard in a random tube Ann. Probab. 38 (2010) 1019-1061. MR 2674993 (2012c:60244)
  • 8. F. Comets, S. Popov, G. Schütz and M. Vachkovskaia, Knudsen gas in a finite random tube: transport diffusion and first passage properties. J. Stat. Phys. 140 (2010) 948-984. MR 2673342 (2012b:60317)
  • 9. S. Evans, Stochastic billiards on general tables. Ann. Appl. Probab. 11, 419-437 (2001). MR 1843052 (2002f:60134)
  • 10. K. Falconer, Fractal geometry. Mathematical foundations and applications. Second edition. John Wiley & Sons, Inc., Hoboken, NJ, 1990. MR 1102677 (92j:28008)
  • 11. R. Feres, Random walks derived from billiards. Dynamics, ergodic theory, and geometry, 179-222, Math. Sci. Res. Inst. Publ., 54, Cambridge Univ. Press, Cambridge, 2007. MR 2369447 (2009c:37033)
  • 12. R. Feres and H.-K. Zhang, The spectrum of the billiard Laplacian of a family of random billiards. J. Stat. Phys. 141, (2010) 1039-1054. MR 2740402 (2011h:37052)
  • 13. R. Klette, K. Schlüns and A. Koschan, Computer vision. Three-dimensional data from images. Springer-Verlag Singapore, Singapore, 1998. MR 1729491 (2000j:68190)
  • 14. S. Lalley and H. Robbins, Stochastic search in a convex region. Probab. Theory Relat. Fields 77, 99-116 (1988). MR 921821 (89f:90078)
  • 15. M. Lapidus and R. Niemeyer, Towards the Koch snowflake fractal billiard: computer experiments and mathematical conjectures. Gems in experimental mathematics, 231-263, Contemp. Math., 517, Amer. Math. Soc., Providence, RI, 2010. MR 2731085 (2012b:37101)
  • 16. M. Lapidus and R. Niemeyer, Families of Periodic Orbits of the Koch Snowflake Fractal Billiard. arXiv:1105.0737
  • 17. S. Tabachnikov, Geometry and billiards. Student Mathematical Library, 30. American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2005. MR 2168892 (2006h:51001)
  • 18. Wikipedia, Retroreflector, Online; accessed 4-March-2012.
  • 19. Wikipedia, Scotchlite, Online; accessed 4-March-2012.

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Additional Information

Omer Angel
Affiliation: Department of Mathematics, University of British Columbia, 121 - 1984 Mathematics Road, Vancouver, British Columbia, Canada V6T 1Z2

Krzysztof Burdzy
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195

Scott Sheffield
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 2-180, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139

Received by editor(s): March 4, 2012
Received by editor(s) in revised form: April 7, 2012, and April 9, 2012
Published electronically: June 3, 2013
Additional Notes: The first author’s research was supported in part by NSERC and by the Sloan Foundation. The second author’s research was supported in part by NSF Grant DMS-0906743 and by grant N N201 397137, MNiSW, Poland. The third author’s research was supported in part by NSF Grant DMS 0645585.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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