List of probabilistic proofs of non-probabilistic theorems

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

• Normal numbers exist. Moreover, computable normal numbers exist. These non-probabilistic existence theorems follow from probabilistic results: (a) a number chosen at random (uniformly on (0,1)) is normal almost surely (which follows easily from the strong law of large numbers); (b) some probabilistic inequalities behind the strong law. The existence of a normal number follows from (a) immediately. The proof of the existence of computable normal numbers, based on (b), involves additional arguments. All known proofs use probabilistic arguments.
• Dvoretzky's theorem which states that high-dimensional convex bodies have ball-like slices is proved probabilistically. No deterministic construction is known, even for many specific bodies.
• The diameter of the Banach–Mazur compactum was calculated using a probabilistic construction. No deterministic construction is known.
• The original proof that the Hausdorff–Young inequality cannot be extended to ${\displaystyle p>2}$ is probabilistic. The proof of the de Leeuw–Kahane–Katznelson theorem (which is a stronger claim) is partially probabilistic.[1]
• The first construction of a Salem set was probabilistic.[2] Only in 1981 did Kaufman give a deterministic construction.[3]
• Every continuous function on a compact interval can be uniformly approximated by polynomials, which is the Weierstrass approximation theorem. A probabilistic proof uses the weak law of large numbers. Non-probabilistic proofs were available earlier.
• Existence of a nowhere differentiable continuous function follows easily from properties of Wiener process. A non-probabilistic proof was available earlier.
• Stirling's formula was first discovered by Abraham de Moivre in his `The Doctrine of Chances' (with a constant identified later by Stirling) in order to be used in probability theory. Several probabilistic proofs of Stirling's formula (and related results) were found in the 20th century.[4][5]
• The only bounded harmonic functions defined on the whole plane are constant functions by Liouville's theorem. A probabilistic proof via two-dimensional Brownian motion is well known.[6] Non-probabilistic proofs were available earlier.
• Non-tangential boundary values[7] of an analytic or harmonic function exist at almost all boundary points of non-tangential boundedness. This result (Privalov's theorem), and several results of this kind, are deduced from martingale convergence.[8] Non-probabilistic proofs were available earlier.
• The boundary Harnack principle is proved using Brownian motion[9] (see also[10]). Non-probabilistic proofs were available earlier.
• Euler's Basel sum, ${\displaystyle \qquad \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}},}$ 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

• A number of theorems stating existence of graphs (and other discrete structures) with desired properties are proved by the probabilistic method. Non-probabilistic proofs are available for a few of them.
• The maximum-minimums identity admits a probabilistic proof.

Topology and geometry

• A smooth boundary is evidently two-sided, but a non-smooth (especially, fractal) boundary can be quite complicated. It was conjectured to be two-sided in the sense that the natural projection of the Martin boundary[15] to the topological boundary is at most 2 to 1 almost everywhere.[16] This conjecture is proved using Brownian motion, local time, stochastic integration, coupling, hypercontractivity etc.[17] (see also[18]). A non-probabilistic proof is found 18 years later.[19]
• The Loewner's torus inequality relates the area of a compact surface (topologically, a torus) to its systole. It can be proved most easily by using the probabilistic notion of variance.[20] A non-probabilistic proof was available earlier.
• The weak halfspace theorem for minimal surfaces states that any complete minimal surface of bounded curvature which is not a plane is not contained in any halfspace. This theorem is proved using a coupling between Brownian motions on minimal surfaces.[21] A non-probabilistic proof was available earlier.

Quantum theory

• Non-commutative dynamics (called also quantum dynamics) is formulated in terms of Von Neumann algebras and continuous tensor products of Hilbert spaces.[23] Several results (for example, a continuum of mutually non-isomorphic models) are obtained by probabilistic means (random compact sets and Brownian motion).[24][25] One part of this theory (so-called type III systems) is translated into the analytic language[26] and is developing analytically;[27] the other part (so-called type II systems) exists still in the probabilistic language only.
• Tripartite quantum states can lead to arbitrary large violations of Bell inequalities[28] (in sharp contrast to the bipartite case). The proof uses random unitary matrices. No other proof is available.

Notes

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: 353–365. Bibcode:1951ArM.....1..353S. doi:10.1007/bf02591372.
3. ^ Kaufman, Robert (1981). "On the theorem of Jarník and Besicovitch". Acta Arith. 39 (3): 265–267.
4. ^ Blyth, Colin R.; Pathak, Pramod K. (1986), "A note on easy proofs of Stirling's theorem", American Mathematical Monthly, 93 (5): 376–379, doi:10.2307/2323600, JSTOR 2323600.
5. ^ Gordon, Louis (1994), "A stochastic approach to the gamma function", American Mathematical Monthly, 101 (9): 858–865, doi:10.2307/2975134, JSTOR 2975134.
6. ^ a b Revuz, Daniel; Yor, Marc (1994), Continuous martingales and Brownian motion (2nd ed.), Springer (see Exercise (2.17) in Section V.2, page 187).
7. ^ See Fatou's theorem.
8. ^ Durrett, Richard (1984), Brownian motion and martingales in analysis, California: Wadsworth, ISBN 0-534-03065-3.
9. ^ Bass, R.F.; Burdzy, K. (1989), "A probabilistic proof of the boundary Harnack principle", Seminar on Stochastic Processes, Boston: Birkhäuser (published 1990), pp. 1–16, hdl:1773/2249.
10. ^ Bass, Richard F. (1995), Probabilistic techniques in analysis, Springer, p. 228.
11. ^ Markowsky, Greg T. (2011), "On the expected exit time of planar Brownian motion from simply connected domains", Electronic Communications in Probability, 16: 652–663, doi:10.1214/ecp.v16-1653.
12. ^ Davis, Burgess (1975), "Picard's theorem and Brownian motion", Transactions of the American Mathematical Society, 213: 353–362, doi:10.2307/1998050.
13. ^ Davis, Burgess (1979), "Brownian motion and analytic functions", Annals of Probability, 7: 913–932, doi:10.1214/aop/1176994888.
14. ^ Bismut, Jean-Michel (1984), "The Atiyah–Singer Theorems: A Probabilistic Approach. I. The index theorem", J. Funct. Analysis, 57: 56–99, doi:10.1016/0022-1236(84)90101-0.
15. ^ As long as we have no article on Martin boundary, see Compactification (mathematics)#Other compactification theories.
16. ^ Bishop, C. (1991), "A characterization of Poissonian domains", Arkiv för Matematik, 29 (1): 1–24, Bibcode:1991ArM....29....1B, doi:10.1007/BF02384328 (see Section 6).
17. ^ Tsirelson, Boris (1997), "Triple points: from non-Brownian filtrations to harmonic measures", Geometric and Functional Analysis, Birkhauser, 7 (6): 1096–1142, doi:10.1007/s000390050038. author's site
18. ^ Tsirelson, Boris (1998), "Within and beyond the reach of Brownian innovation", Proceedings of the international congress of mathematicians, Documenta mathematica, Extra Volume ICM 1998, III, Berlin: der Deutschen Mathematiker-Vereinigung, pp. 311–320, ISSN 1431-0635.
19. ^ Tolsa, Xavier; Volberg, Alexander (2017). "On Tsirelson's theorem about triple points for harmonic measure". International Mathematics Research Notices. arXiv:1608.04022. doi:10.1093/imrn/rnw345.
20. ^ Horowitz, Charles; Usadi Katz, Karin; Katz, Mikhail G. (2008). "Loewner's torus inequality with isosystolic defect". Journal of Geometric Analysis. 19 (4): 796–808. arXiv:0803.0690. doi:10.1007/s12220-009-9090-y.
21. ^ Neel, Robert W. (2008), "A martingale approach to minimal surfaces", Journal of Functional Analysis, Elsevier, 256 (8): 2440–2472, doi:10.1016/j.jfa.2008.06.033. Also arXiv:0805.0556.
22. ^ Fulman, Jason (2001), "A probabilistic proof of the Rogers–Ramanujan identities", Bulletin of the London Mathematical Society, 33 (4): 397–407, arXiv:math/0001078, doi:10.1017/S0024609301008207. Also arXiv:math.CO/0001078.
23. ^ Arveson, William (2003), Noncommutative dynamics and E-semigroups, New York: Springer, ISBN 0-387-00151-4.
24. ^ Tsirelson, Boris (2003), "Non-isomorphic product systems", in Price, Geoffrey, Advances in quantum dynamics, Contemporary mathematics, 335, American mathematical society, pp. 273–328, ISBN 0-8218-3215-8. Also arXiv:math.FA/0210457.
25. ^ Tsirelson, Boris (2008), "On automorphisms of type II Arveson systems (probabilistic approach)", New York Journal of Mathematics, 14: 539–576.
26. ^ Bhat, B.V.Rajarama; Srinivasan, Raman (2005), "On product systems arising from sum systems", Infinite Dimensional Analysis, Quantum Probability and Related Topics (IDAQP), 8 (1): 1–31, doi:10.1142/S0219025705001834. Also arXiv:math.OA/0405276.
27. ^ Izumi, Masaki; Srinivasan, Raman (2008), "Generalized CCR flows", Communications in Mathematical Physics, 281 (2): 529–571, arXiv:0705.3280, Bibcode:2008CMaPh.281..529I, doi:10.1007/s00220-008-0447-z. Also arXiv:0705.3280.
28. ^ Perez-Garcia, D.; Wolf, M.M.; C., Palazuelos; Villanueva, I.; Junge, M. (2008), "Unbounded violation of tripartite Bell inequalities", Communications in Mathematical Physics, Springer, 279 (2): 455–486, arXiv:quant-ph/0702189, Bibcode:2008CMaPh.279..455P, doi:10.1007/s00220-008-0418-4