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.
- Richard F. Bass, Krzysztof Burdzy, Zhen-Qing Chen, and Martin Hairer, Stationary distributions for diffusions with inert drift, Probab. Theory Related Fields 146 (2010), no. 1-2, 1–47. MR 2550357, DOI 10.1007/s00440-008-0182-6
- S. Chandrasekhar, Radiative transfer, Dover Publications, Inc., New York, 1960. MR 0111583
- Nikolai Chernov and Roberto Markarian, Chaotic billiards, Mathematical Surveys and Monographs, vol. 127, American Mathematical Society, Providence, RI, 2006. MR 2229799, DOI 10.1090/surv/127
- 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).
- Francis Comets, Serguei Popov, Gunter M. Schütz, and Marina Vachkovskaia, Billiards in a general domain with random reflections, Arch. Ration. Mech. Anal. 191 (2009), no. 3, 497–537. MR 2481068, DOI 10.1007/s00205-008-0120-x
- Francis Comets, Serguei Popov, Gunter M. Schütz, and Marina Vachkovskaia, Erratum: Billiards in a general domain with random reflections [MR2481068], Arch. Ration. Mech. Anal. 193 (2009), no. 3, 737–738. MR 2525117, DOI 10.1007/s00205-009-0236-7
- Francis Comets, Serguei Popov, Gunter M. Schütz, and Marina Vachkovskaia, Quenched invariance principle for the Knudsen stochastic billiard in a random tube, Ann. Probab. 38 (2010), no. 3, 1019–1061. MR 2674993, DOI 10.1214/09-AOP504
- Francis Comets, Serguei Popov, Gunter M. Schütz, and Marina Vachkovskaia, Knudsen gas in a finite random tube: transport diffusion and first passage properties, J. Stat. Phys. 140 (2010), no. 5, 948–984. MR 2673342, DOI 10.1007/s10955-010-0023-8
- Steven N. Evans, Stochastic billiards on general tables, Ann. Appl. Probab. 11 (2001), no. 2, 419–437. MR 1843052, DOI 10.1214/aoap/1015345298
- Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR 1102677
- Renato Feres, Random walks derived from billiards, Dynamics, ergodic theory, and geometry, Math. Sci. Res. Inst. Publ., vol. 54, Cambridge Univ. Press, Cambridge, 2007, pp. 179–222. MR 2369447, DOI 10.1017/CBO9780511755187.008
- Renato Feres and Hong-Kun Zhang, The spectrum of the billiard Laplacian of a family of random billiards, J. Stat. Phys. 141 (2010), no. 6, 1039–1054. MR 2740402, DOI 10.1007/s10955-010-0079-5
- Reinhard Klette, Karsten Schlüns, and Andreas Koschan, Computer vision, Springer-Verlag Singapore, Singapore, 1998. Three-dimensional data from images; Translated from the German; Revised by the authors. MR 1729491
- Steven Lalley and Herbert Robbins, Stochastic search in a convex region, Probab. Theory Related Fields 77 (1988), no. 1, 99–116. MR 921821, DOI 10.1007/BF01848133
- Michel L. Lapidus and Robert G. Niemeyer, Towards the Koch snowflake fractal billiard: computer experiments and mathematical conjectures, Gems in experimental mathematics, Contemp. Math., vol. 517, Amer. Math. Soc., Providence, RI, 2010, pp. 231–263. MR 2731085, DOI 10.1090/conm/517/10144
- M. Lapidus and R. Niemeyer, Families of Periodic Orbits of the Koch Snowflake Fractal Billiard. arXiv:1105.0737
- Serge Tabachnikov, Geometry and billiards, Student Mathematical Library, vol. 30, American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2005. MR 2168892, DOI 10.1090/stml/030
- Wikipedia, Retroreflector, http://en.wikipedia.org/wiki/Retroreflector Online; accessed 4-March-2012.
- Wikipedia, Scotchlite, http://en.wikipedia.org/wiki/Scotchlite Online; accessed 4-March-2012.
- Omer Angel
- Affiliation: Department of Mathematics, University of British Columbia, 121 - 1984 Mathematics Road, Vancouver, British Columbia, Canada V6T 1Z2
- MR Author ID: 667585
- Email: firstname.lastname@example.org
- Krzysztof Burdzy
- Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
- Email: email@example.com
- Scott Sheffield
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 2-180, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
- Email: firstname.lastname@example.org
- 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.
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
- Journal: Trans. Amer. Math. Soc. 365 (2013), 6367-6383
- MSC (2010): Primary 37D50
- DOI: https://doi.org/10.1090/S0002-9947-2013-05851-5
- MathSciNet review: 3105755