List of probabilistic proofs of non-probabilistic theorems explained

Probability theory routinely uses results from other fields of mathematics (mostly, analysis). The opposite cases, collected below, are relatively rare; however, probability theory is used systematically in combinatorics via the probabilistic method. They are particularly used for non-constructive proofs.

Analysis

p>2

is probabilistic. The proof of the de Leeuw–Kahane–Katznelson theorem (which is a stronger claim) is partially probabilistic.[1]

  

infin
\sum
n=1
1
n2

=

\pi2
6

,

can be demonstrated by considering the expected exit time of planar Brownian motion from an infinite strip. A number of other less well-known identities can be deduced in a similar manner.[11]

Combinatorics

Algebra

Topology and geometry

Number theory

Quantum theory

Information theory

See also

External links

Notes and References

  1. Karel de Leeuw, Yitzhak Katznelson and Jean-Pierre Kahane, Sur les coefficients de Fourier des fonctions continues. (French) C. R. Acad. Sci. Paris Sér. A–B 285:16 (1977), A1001–A1003.
  2. Salem . Raphaël . 1951 . On singular monotonic functions whose spectrum has a given Hausdorff dimension . Ark. Mat. . 1 . 4. 353–365 . 10.1007/bf02591372. 1951ArM.....1..353S. free .
  3. Kaufman . Robert . 1981 . On the theorem of Jarník and Besicovitch . Acta Arith . 39 . 3. 265–267 . 10.4064/aa-39-3-265-267 . free .
  4. .
  5. .
  6. (see Exercise (2.17) in Section V.2, page 187).
  7. See Fatou's theorem.
  8. .
  9. .
  10. .
  11. .
  12. .
  13. .
  14. .
  15. Bernoulli actions are weakly contained in any free action . 1103.1063v2 . Abért . Miklós . Weiss . Benjamin . 2011 . math.DS .
  16. .
  17. As long as we have no article on Martin boundary, see Compactification (mathematics)#Other compactification theories.
  18. (see Section 6).
  19. . author's site
  20. .
  21. Tolsa . Xavier . Volberg . Alexander . 2017 . On Tsirelson's theorem about triple points for harmonic measure . 1608.04022 . International Mathematics Research Notices . 2018 . 12 . 3671–3683 . 10.1093/imrn/rnw345.
  22. Horowitz . Charles . Mikhail Katz . Usadi Katz . Karin . Katz . Mikhail G. . 2008 . Loewner's torus inequality with isosystolic defect . 0803.0690 . Journal of Geometric Analysis . 19 . 4. 796–808 . 10.1007/s12220-009-9090-y. 18444111 .
  23. . Also arXiv:0805.0556.
  24. . Also arXiv:math.CO/0001078.
  25. .
  26. . Also arXiv:math.FA/0210457.
  27. .
  28. . Also arXiv:math.OA/0405276.
  29. . Also arXiv:0705.3280.