## Deterministic approximations of random reflectors

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- by Omer Angel, Krzysztof Burdzy and Scott Sheffield PDF
- Trans. Amer. Math. Soc.
**365**(2013), 6367-6383 Request permission

## 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

- 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.

## Additional Information

**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: angel@math.ubc.ca
**Krzysztof Burdzy**- Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
- Email: burdzy@math.washington.edu
**Scott Sheffield**- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 2-180, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
- Email: sheffield@math.mit.edu
- 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