55:); (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.
266:
500:). One part of this theory (so-called type III systems) is translated into the analytic language and is developing analytically; the other part (so-called type II systems) exists still in the probabilistic language only.
144:' (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.
268:
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.
19:
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
557:
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.
51:. 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
379:
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
61:
which states that high-dimensional convex bodies have ball-like slices is proved probabilistically. No deterministic construction is known, even for many specific bodies.
99:
192:
173:
429:
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
344:
which is a lower bound on the number of crossing for any drawing of a graph as a function of the number of vertices, edges the graph has.
314:
1529:
Perez-Garcia, D.; Wolf, M.M.; C., Palazuelos; Villanueva, I.; Junge, M. (2008), "Unbounded violation of tripartite Bell inequalities",
1376:
1348:
1279:
755:
411:
403:
108:
1607:
1124:, Documenta mathematica, vol. Extra Volume ICM 1998, III, Berlin: der Deutschen Mathematiker-Vereinigung, pp. 311–320,
489:
467:
362:
72:
1024:
355:
101:
is probabilistic. The proof of the de Leeuw–Kahane–Katznelson theorem (which is a stronger claim) is partially probabilistic.
365:
is proved using probabilistic methods (rather than heat equation methods). A non-probabilistic proof was available earlier.
327:
A number of theorems stating existence of graphs (and other discrete structures) with desired properties are proved by the
415:
1602:
522:
52:
65:
492:. Several results (for example, a continuum of mutually non-isomorphic models) are obtained by probabilistic means (
104:
The first construction of a Salem set was probabilistic. Only in 1981 did
Kaufman give a deterministic construction.
341:
1119:
335:
292:
116:
180:
507:(in sharp contrast to the bipartite case). The proof uses random unitary matrices. No other proof is available.
388:
1143:
Tolsa, Xavier; Volberg, Alexander (2017). "On
Tsirelson's theorem about triple points for harmonic measure".
455:, providing the first proof of existence of normal numbers, with the help of the first version of the strong
141:
107:
Every continuous function on a compact interval can be uniformly approximated by polynomials, which is the
396:
285:
58:
133:
1548:
1491:
1046:
580:
539:
456:
452:
358:
can be proved using two-dimensional
Brownian motion. Non-probabilistic proofs were available earlier.
328:
310:
161:
127:
112:
24:
967:
Abért, Miklós; Weiss, Benjamin (2011). "Bernoulli actions are weakly contained in any free action".
485:
392:
376:
298:
1564:
1538:
1507:
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1450:
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1309:
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1094:
988:
968:
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847:
829:
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658:
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383:
to the topological boundary is at most 2 to 1 almost everywhere. This conjecture is proved using
281:
16:
929:
730:
1317:
1372:
1344:
1125:
751:
504:
433:
between
Brownian motions on minimal surfaces. A non-probabilistic proof was available earlier.
165:
150:
137:
44:
1423:
Bhat, B.V.Rajarama; Srinivasan, Raman (2005), "On product systems arising from sum systems",
1556:
1499:
1442:
1371:, Contemporary mathematics, vol. 335, American mathematical society, pp. 273–328,
1301:
1248:
1203:
1162:
1086:
1054:
1000:
941:
909:
874:
839:
779:
685:
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621:
588:
526:
183:
is proved using
Brownian motion (see also). Non-probabilistic proofs were available earlier.
122:
Existence of a nowhere differentiable continuous function follows easily from properties of
78:
28:
1103:
1396:
1364:
1336:
1115:
1074:
1020:
797:
771:
641:
Blyth, Colin R.; Pathak, Pramod K. (1986), "A note on easy proofs of
Stirling's theorem",
426:
407:
380:
306:
154:
68:
was calculated using a probabilistic construction. No deterministic construction is known.
168:
function exist at almost all boundary points of non-tangential boundedness. This result (
149:
The only bounded harmonic functions defined on the whole plane are constant functions by
1552:
1495:
1050:
584:
399:, hypercontractivity etc. (see also). A non-probabilistic proof is found 18 years later.
518:
497:
448:
384:
302:
274:
123:
991:(1984), "The Atiyah–Singer Theorems: A Probabilistic Approach. I. The index theorem",
1596:
1098:
1005:
953:
444:
261:{\displaystyle \qquad \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}},}
186:
169:
48:
40:
20:
1568:
1511:
1454:
1260:
1215:
851:
1587:
1181:
818:"On the expected exit time of planar Brownian motion from simply connected domains"
743:
471:
1313:
484:
Non-commutative dynamics (called also quantum dynamics) is formulated in terms of
774:; Burdzy, K. (1989), "A probabilistic proof of the boundary Harnack principle",
569:"On singular monotonic functions whose spectrum has a given Hausdorff dimension"
1560:
1503:
1446:
1305:
1252:
1207:
945:
1425:
Infinite
Dimensional Analysis, Quantum Probability and Related Topics (IDAQP)
1129:
1077:(1997), "Triple points: from non-Brownian filtrations to harmonic measures",
932:; Yam, S.C.P. (2018), "A probabilistic proof for Fourier inversion formula",
914:
626:
609:
1400:
1166:
1090:
817:
1543:
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284:
on the real line is differentiable almost everywhere can be proved using
1059:
886:
697:
662:
593:
568:
1401:"On automorphisms of type II Arveson systems (probabilistic approach)"
317:
action of a discrete, countable group on a standard probability space.
1437:
1296:
1229:
Neel, Robert W. (2008), "A martingale approach to minimal surfaces",
783:
676:
Gordon, Louis (1994), "A stochastic approach to the gamma function",
291:
Multidimensional
Fourier inversion formula can be established by the
277:
can be proved using the winding properties of planar Brownian motion.
1367:(2003), "Non-isomorphic product systems", in Price, Geoffrey (ed.),
973:
878:
689:
654:
503:
Tripartite quantum states can lead to arbitrary large violations of
1157:
1583:
1516:
1486:
1472:
Izumi, Masaki; Srinivasan, Raman (2008), "Generalized CCR flows",
1265:
1243:
1198:
834:
900:
Davis, Burgess (1979), "Brownian motion and analytic functions",
525:
uses random coding to show the existence of a code that achieves
172:'s theorem), and several results of this kind, are deduced from
1037:
Bishop, C. (1991), "A characterization of Poissonian domains",
865:
Davis, Burgess (1975), "Picard's theorem and Brownian motion",
157:
is well known. Non-probabilistic proofs were available earlier.
1184:(2008). "Loewner's torus inequality with isosystolic defect".
1118:(1998), "Within and beyond the reach of Brownian innovation",
1025:
Compactification (mathematics)#Other compactification theories
1121:
Proceedings of the international congress of mathematicians
331:. Non-probabilistic proofs are available for a few of them.
1280:"A probabilistic proof of the Rogers–Ramanujan identities"
778:, Boston: Birkhäuser (published 1990), pp. 1–16,
195:
81:
474:. A non-probabilistic proof was available earlier.
422:. A non-probabilistic proof was available earlier.
295:and some elementary results from complex analysis.
260:
176:. Non-probabilistic proofs were available earlier.
119:. Non-probabilistic proofs were available earlier.
93:
867:Transactions of the American Mathematical Society
720:(see Exercise (2.17) in Section V.2, page 187).
305:proved, via a probabilistic construction, that
8:
451:, could be one of the first examples of the
1284:Bulletin of the London Mathematical Society
748:Brownian motion and martingales in analysis
1145:International Mathematics Research Notices
716:Continuous martingales and Brownian motion
610:"On the theorem of JarnĂk and Besicovitch"
153:. A probabilistic proof via n-dimensional
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1341:Noncommutative dynamics and E-semigroups
822:Electronic Communications in Probability
459:(see also the first item of the section
550:
1584:Probabilistic Proofs of Analytic Facts
1531:Communications in Mathematical Physics
1474:Communications in Mathematical Physics
1180:Horowitz, Charles; Usadi Katz, Karin;
309:are weakly contained (in the sense of
418:by using the probabilistic notion of
160:Non-tangential boundary values of an
7:
934:Statistics & Probability Letters
802:Probabilistic techniques in analysis
709:
707:
363:index theorem for elliptic complexes
213:
14:
1079:Geometric and Functional Analysis
1019:As long as we have no article on
714:Revuz, Daniel; Yor, Marc (1994),
490:tensor products of Hilbert spaces
109:Weierstrass approximation theorem
27:. They are particularly used for
410:(topologically, a torus) to its
1405:New York Journal of Mathematics
776:Seminar on Stochastic Processes
425:The weak halfspace theorem for
196:
1231:Journal of Functional Analysis
356:fundamental theorem of algebra
1:
1186:Journal of Geometric Analysis
678:American Mathematical Monthly
643:American Mathematical Monthly
460:
338:admits a probabilistic proof.
1369:Advances in quantum dynamics
1085:(6), Birkhauser: 1096–1142,
1006:10.1016/0022-1236(84)90101-0
71:The original proof that the
816:Markowsky, Greg T. (2011),
468:Rogers–Ramanujan identities
53:strong law of large numbers
1624:
1237:(8), Elsevier: 2440–2472,
404:Loewner's torus inequality
342:Crossing number inequality
181:boundary Harnack principle
73:Hausdorff–Young inequality
1561:10.1007/s00220-008-0418-4
1504:10.1007/s00220-008-0447-z
1447:10.1142/S0219025705001834
1306:10.1017/S0024609301008207
1253:10.1016/j.jfa.2008.06.033
1208:10.1007/s12220-009-9090-y
946:10.1016/j.spl.2018.05.028
750:, California: Wadsworth,
336:maximum-minimums identity
315:measure-preserving action
293:weak law of large numbers
117:weak law of large numbers
718:(2nd ed.), Springer
608:Kaufman, Robert (1981).
136:was first discovered by
1608:Probabilistic arguments
804:, Springer, p. 228
627:10.4064/aa-39-3-265-267
567:Salem, Raphaël (1951).
447:theorem (1909), due to
142:The Doctrine of Chances
128:non-probabilistic proof
1343:, New York: Springer,
1278:Fulman, Jason (2001),
915:10.1214/aop/1176994888
523:channel coding theorem
406:relates the area of a
393:stochastic integration
286:martingale convergence
262:
217:
174:martingale convergence
130:was available earlier.
95:
94:{\displaystyle p>2}
75:cannot be extended to
66:Banach–Mazur compactum
1460:arXiv:math.OA/0405276
1384:arXiv:math.FA/0210457
1324:arXiv:math.CO/0001078
1091:10.1007/s000390050038
902:Annals of Probability
370:Topology and geometry
263:
197:
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844:10.1214/ecp.v16-1653
540:Probabilistic method
486:Von Neumann algebras
457:law of large numbers
453:probabilistic method
329:probabilistic method
280:The fact that every
193:
79:
64:The diameter of the
25:probabilistic method
1603:Mathematical proofs
1553:2008CMaPh.279..455P
1496:2008CMaPh.281..529I
1167:10.1093/imrn/rnw345
1051:1991ArM....29....1B
1039:Arkiv för Matematik
989:Bismut, Jean-Michel
585:1951ArM.....1..353S
494:random compact sets
151:Liouville's theorem
113:probabilistic proof
59:Dvoretzky's theorem
1060:10.1007/BF02384328
594:10.1007/bf02591372
512:Information theory
416:proved most easily
282:Lipschitz function
258:
134:Stirling's formula
91:
17:Probability theory
1151:(12): 3671–3683.
505:Bell inequalities
470:are proved using
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187:Euler's Basel sum
138:Abraham de Moivre
43:exist. Moreover,
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1045:(1): 1–24,
974:1103.1063v2
940:: 135–142,
873:: 353–362,
828:: 652–663,
449:Émile Borel
1597:Categories
1158:1608.04022
930:Wong, T.K.
772:Bass, R.F.
614:Acta Arith
389:local time
45:computable
1487:0705.3280
1411:: 539–576
1244:0805.0556
1199:0803.0690
1130:1431-0635
1099:121617197
999:: 56–99,
954:125351871
835:1108.1188
784:1773/2249
375:A smooth
242:π
214:∞
199:∑
115:uses the
1569:29110154
1512:12815055
1455:15106610
1399:(2008),
1339:(2003),
1261:15228691
1216:18444111
852:55705658
800:(1995),
746:(1984),
573:Ark. Mat
534:See also
461:Analysis
431:coupling
420:variance
397:coupling
377:boundary
170:Privalov
166:harmonic
162:analytic
140:in his `
35:Analysis
31:proofs.
23:via the
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412:systole
349:Algebra
311:Kechris
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1292:arXiv
1257:S2CID
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1194:arXiv
1153:arXiv
1095:S2CID
969:arXiv
950:S2CID
883:JSTOR
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830:arXiv
694:JSTOR
659:JSTOR
546:Notes
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299:Abért
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