Abstract
The book’s introduction presents the main mathematical themes considered by Paul Lévy and Maurice Fréchet in their correspondence to one another and examines the scientific and institutional context in which their letters were exchanged during their nearly fifty years of correspondence.
The book is divided into helpful sections. A first section is devoted to a short presentation of Emile Borel and Jacques Hadamard, who were mentors to Lévy and Fréchet. The second section examines the probabilistic stage in France at the turn of the century, during which time Lévy and Fréchet were students. The third section studies several aspects of the probabilistic work in Russia and Soviet Union.
The authors have provided information on how Soviet Union became the center for the study of probability theory between the two world wars. The authors also examine the similarities between Lévy’s and Fréchet’s interests and the discovery of Lévy’s stable distributions.
Finally, three sections concentrate on the history of three major topics of Lévy’s studies in probability theory: potential theory, Brownian motion and stochastic integration.
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- 1.
(1890) Acta Math. 13:1–270.
- 2.
- 3.
(1905) Bull. Soc. Math. Fr. 33:123–128.
- 4.
(1909) Rend. Circ. Mat. Palermo 27:247–270.
- 5.
Kolmogoroff, Andrei (1931) Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 104:415–458.
- 6.
Our emphasis.
- 7.
Petrograd is the name taken by Saint Petersburg from the beginning of the First World War until 1924 when it became Leningrad.
- 8.
Seneta (2001) affirms that this occurred in reaction to an article by Nekrassov, explaining that a proof of the existence of free will emerges from the everyday observation of averaging phenomenon in society (for example, the average number of births remains almost the same from one year to the next) because independence of actions is necessary to apply the law of large numbers. Markov, extremely anti-religious, was eager to disprove this necessity.
- 9.
Luzin and Egorov had close ties to orthodox theological circles, notably to the extraordinary Pavel Alexandrovitch Florensky (1882–1937), Luzin’s close friend, who managed both studies in mathematics at the University of Moscow and religion at the Academy of Theology. In 1912, at the Academy, he defended a thesis searching for, à la V. Soloviev, a spiritual interpretation of science. Luzin also spent some time studying theology. The correspondence between Luzin and Florensky is the object of an interesting paper (Демидов et al. 1989). On Florensky, see the beautiful book Betti (2010), and also Žust (2002) for biographical aspects.
- 10.
Wordplay on the name Luzin, probably an allusion to the Lusitania, sunk in 1916, which was a pretext for the USA to enter the First World War.
- 11.
Detailed information about Slutsky is available in Locker (2001), as well as in the 1948 eulogy to him written by Kolmogorov (Колмогоров А.Н. (1948) Е.Е. Слуцкий. Успехи Математической Наук III(4):142–151).
- 12.
See, for instance, the illuminating resume of the XVIIIth Congress of the Party published as an introduction to the 1939 volume of the Известия Академии Наук СССР.
- 13.
- 14.
See Демидов and Левшин (1999).
- 15.
Which, most ironically, is also the reason for its lack of esteem by the French “structuralist” school (see Maashal 1999).
- 16.
The archives of the Institut Henri Poincaré also contain an enormous dossier on the physicist Frenkel, who postponed his visit from 1934 to 1937.
- 17.
Draft of a speech in Kiev, archives of the Paris Academy of Science.
- 18.
On Kolmogorov, the reader is invited to consult (Chaumont et al. 2007).
- 19.
Kolmogoroff, Andrei (1931) Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 104:415–458.
- 20.
See Chap. 7 of Von Plato (1994) on this question.
- 21.
Letter, August 3, 1939 (archives of the Paris Academy of Science). The very late date of this letter, from a moment when contacts between Soviet and western scientists had almost disappeared, is another sign of the benevolent attitude of the regime toward Kolmogorov.
- 22.
For example, during the winter of 1933–1934, Kolmogorov wrote to Fréchet (January 29, 1934, archives of the Paris Academy of Science) that he had obtained the criterion for a process in continuous time in the form E(|f(x+Δ)−f(x)|K)=O(|Δ|1+ε), which he applied to Brownian motion. Oddly, the famous criterion that bears his name was never published by Kolmogorov himself, but appears for the first time (properly attributed to him) in an article by Slutsky in 1937. Incidentally, the letter of Kolmogorov just mentioned is written in an uncharacteristically anxious style. He wrote for instance: “It requires considerable energy to surmount formal difficulties” suggesting that he was asking Borel for some official help so that he could obtain the permission to travel). Kolmogorov apparently encountered unexpected bureaucratic difficulties regarding his trip to France, a sign of the restrictions that the Soviet authorities were imposing on international travel.
- 23.
(1910) Rend. Circ. Mat. Palermo 30:1–26.
- 24.
On Gateaux, see the complete study provided in Mazliak (2011).
- 25.
(1992) Vrin, Paris.
- 26.
On Daniell, see Aldrich (2007).
- 27.
On this important topic, see the many details contained in the paper of Mazliak (2011).
- 28.
Lévy, Paul (1919) Sur la notion de moyenne dans le domaine fonctionnel. CRAS (August 25, 1919).
- 29.
Balanzat, Manuel (1960) La différentielle d’Hadamard-Fréchet dans les espaces vectoriels topologiques. CRAS (November 28, 1960).
- 30.
For details on this important moment in Fréchet’s life, see Havlova et al. (2005).
- 31.
For Borel’s interest in probability, see Durand and Mazliak (2011).
- 32.
Presses Universitaires de France, Paris (1955).
- 33.
See Bru (1999).
- 34.
Cramer, Harald (1958) Eloge de M. Paul Lévy. In: Le calcul des probabilités et ses applications (July 15–20, 1958). Editions du CNRS, Paris. pp. 13–15.
- 35.
See Barbut and Mazliak (2008).
- 36.
(1923) Математический Сборник 31:296–301.
- 37.
Czuber, Emanuel (1891) Theorie des Beobachtungsfehler. Teubner, Leipzig.
- 38.
(1924) Математический Сборник 32:5–8.
- 39.
(1925) CRAS 180:1716–1719.
- 40.
Fréchet, Maurice (1925) Sur la loi des erreurs d’observation. CRAS 181:204–205.
- 41.
Fréchet, Maurice (1928) Sur la loi de probabilités de l’écart maximum. Ann. Soc. Pol. Math. 6:93–122 and Fréchet, Maurice (1928) Sur l’hypothèse de l’additivité des erreurs partielles. Bull. Soc. Math. Fr. 52:203–216.
- 42.
Lévy, Paul (1929) Sur quelques travaux relatifs à la théorie des erreurs. Bull. Soc. Math. Fr. 53:11–32.
- 43.
See Fréchet and Halbwachs (1924, Chap. V).
- 44.
See Mazliak (2014).
- 45.
In the complementary notes to the second edition of Lévy (1937) Lévy mentions that Fréchet concurred.
- 46.
Lévy (1937), note II in the second edition of 1954.
- 47.
French: type de loi.
- 48.
Or an average, or more generally a positive linear combination.
- 49.
CRAS (February 3, 1936).
- 50.
- 51.
One can read about Lhoste in Hadjadji Seddik-Ameur, Nacira (2003) Les tests de normalité de Lhoste. Math. Sci. Hum. 162:19–43.
- 52.
Lévy, Paul (1922) Sur le rôle de la loi de Gauss dans la théorie des erreurs. CRAS 174:855–857.
- 53.
(1924) Bull. Soc. Math. Fr. 52:49–85.
- 54.
Ann. Soc. Pol. Math. 5:93–116.
- 55.
(1929) Bull. Soc. Math. Fr. 53:1–21.
- 56.
Intégrales à éléments aléatoires indépendants et lois stables à n variables. CRAS (March 17, 1936).
- 57.
Pareto, Vilfredo. Cours d’Économie Politique Professé a l’Université de Lausanne. Volume I (1896), Volume II (1897). F. Rouge, Lausanne.
- 58.
See the note 6 to the Preface by Kaï Laï Chung.
- 59.
- 60.
- 61.
See footnote 62 below.
- 62.
The double layer potential consists of a surface distribution of dipole moments all oriented toward the same side of a double-sided surface. The double layer potential may also be viewed as the result of two surface charge distributions when the surfaces are infinitesimally close. The usage of the word “distribution” here follows that of physics of the period, that is as a density of electric charges. During the first quarter of the 20th century this was a continuous function. Only later, in the work of Schwartz, did this concept as well as that of a unit-doublet find definitions not requiring a limit by means of distribution theory.
- 63.
This refers to the Berlin academic Carl Neumann (1832–1925), who was editor of Mathematische Annalen.
- 64.
In two dimensions the potential is logarithmic and the singularity which must be removed is lnPM=lnr, rather than 1/PM=1/r.
- 65.
Independently of Green, Gauss’ mean-value theorem for harmonic functions also shows that the potential is determined by its values on the boundary. This result allows one to derive the Poisson representation formula.
- 66.
This designation is used deliberately, since it was introduced by Lévy and employed in 1926 by Bouligand in the volume of the Mémorial des Sciences Mathématiques he devoted to these questions.
- 67.
Especially for the question of the theorem on the conformal representation in the complex plane which is related to the questions considered here. It is well known that in two dimensions, potential theory has a close relation to the theory of holomorphic functions.
- 68.
The Green function is itself the solution of a Dirichlet problem. Except for domains with a boundary consisting of curves or surfaces of a simple geometry, for which the Green function is known from the work of Poisson, Liouville, Cauchy and their successors, very few explicit examples are available. However numerical methods may be used to determine Green functions to a high degree of accuracy.
- 69.
Lévy, Paul (1909) Sur les valeurs de la fonction de Green dans le voisinage du contour. Bull. Soc. Math. Fr. 34:186–190.
- 70.
Lévy, Paul (1911) Une généralisation de la méthode de Fredholm pour la résolution du problème de Dirichlet. J. Ec. Polytech. 15:197–210. Fredholm, the Swedish creator of the theory of integral equations that bears his name, developed this method in 1900.
- 71.
He called Riemann’s method (which uses Dirichlet’s principle) ‘simple and elegant but not rigorous,’ and the other methods are quite insufficient in Poincaré’s opinion for solving Dirichlet’s problem.
- 72.
Hilbert declared at the congress of 1900: ‘Every problem in the calculus of variations has a solution provided that one makes the appropriate hypotheses on the boundary conditions and, if necessary sufficiently extends the concept of solution’. However in 1901, in order to give a proper proof of Riemann’s theorem, he had to substitute the calculus of variations for very sophisticated theorems on equicontinuous families of harmonic functions.
- 73.
Lévy (1970, p. 42).
- 74.
Lévy, Paul (1935) Sur une forme tensorielle des équations fonctionnelles des fonctions de Green et de Neumann. CRAS 200:1723–1725.
- 75.
Astérique (1988) n∘157–158.
- 76.
See the article Fichera, Gaetano (1994) Vito Volterra and the birth of functional analysis. In Pier (1994), pp. 171–183.
- 77.
For this fascinating story, see the detailed study of Mazliak (2011).
- 78.
For the classic functionals of mechanics of the form \(\int_{0}^{1} f(t,x,\dot{x})\,dt\), the variation cannot be put in this form, except in certain special cases. For linear functionals, which were characterized in 1903 by Hadamard and transformed in 1907 by Riesz into a representation as Stieltjès integrals \(\int_{0}^{1} x(t)\,dF(t)\), the variation can again not be put in Volterra’s form, because of the possibility of points of discontinuity of F. Also, Volterra’s form is not possible when U(x)=x(τ) (i.e. when F is the Dirac measure at the point τ) for which δU=(δx)(τ).
- 79.
Wiener, Norbert (1924) Une condition nécessaire et suffisante de possibilité pour le problème de Dirichlet. CRAS 178:1050–1053.
- 80.
Lebesgue discovered a famous counter-example of insolvability of the Dirichlet problem (called Lebesgue’s spine). This demonstrates the close relation between the existence of a solution and the topology of the boundary. He was preceded in this type of example by the Polish mathematician Zaremba.
- 81.
Lebesgue is referring to the two notes in CRAS by Bouligand on November 3 and 17, 1919.
- 82.
Wiener, Norbert (1925) Note on a paper of O. Perron. J. Math. Phys. 4:21–32.
- 83.
CRAS (November 26, 1945).
- 84.
This version of the theorem is sufficient to understand the Lévy-Fréchet correspondence. Modern definitions of the harmonic measure are more subtle and their connection with the Dirichlet problem more general. This more recent work was originated by Wiener and then extended by Brelot. The harmonic measure appears as a Radon measure H. If f is super-harmonic in a given open set and F is the set of super-harmonic functions g such that lim g(x)≥f(y) (for x an interior point and y on the boundary), one may consider \(\overline{H}(f) = \inf F \) and \(\underline{H}(f) = -\overline{H}(-f)\). When \(\underline{H}(f) =\overline{H}(f)=H(f)\) the boundary condition f is resolutive, meaning that H(f) is a generalized solution of the Dirichlet problem. Wiener had already shown that f↦H(f)(x) is a Radon measure H x . This Radon measure is nothing but the functional representation of the measure. In 1945 Marcel Brelot considered the compactified projective space of a real vector space of arbitrary finite dimension, and redefined the ramified harmonic measure by an extension of the previous method to the space, making use of Daniell’s integral.
- 85.
CRAS (May 25, 1970).
- 86.
Chapters XII–XVI, p. 340, revised edition of 1987.
- 87.
An analysis of Gouy’s work and its importance for physics appeared in 1941 (Picard 1941).
- 88.
See the discussion in Sect. 2.
- 89.
At this time Einstein also revolutionized mechanics with his theory of special relativity and reintroduced the particle picture of light, i.e. photons and the photoelectric effect.
- 90.
- 91.
For details on this transition from Gateaux and Lévy’s works to Wiener’s, see Mazliak (2011).
- 92.
Wiener, Norbert (1924) Un problème de probabilités dénombrables. Bull. Soc. Math. Fr. 52:569–578.
- 93.
Lévy, Paul (1934) Les généralisations de l’espace différentiel de N. Wiener. CRAS (February 26); Lévy, Paul (1934) Sur les espaces V et W. CRAS (March 26); and Lévy, Paul (1934) Complément à l’étude des espaces V et W. CRAS (May 7).
- 94.
The correspondence makes clear that when using the Borel-Cantelli lemma, Lévy always emphasized the advantage of Borel’s result over Cantelli’s in the independent case.
- 95.
In his important note Slutsky, Eugène (1928) Sur les fonctionnelles éventuelles continues et dérivables dans le sens stochastique. CRAS 187:878–880, Slutsky defined and constructed “regular random functions,” i.e. the processes called second order today. He showed that a quadratic mean existed as well as an integral which he wrote as \((S)\int_{a}^{b} x_{t}\,dt\) and called the stochastic integral.
- 96.
Wiener, Norbert (1933). Math. Z. 37:647–668.
- 97.
Wiener, Norbert (1934) Colloq. Publ.-Am. Math. Soc. XIX. Most of this text was written with Raymond Paley who died in a skiing accident in 1933.
- 98.
See Sect. I.8, p. 43 ff, Naive Stochastic Integration is Impossible in Protter (1990).
- 99.
On this topic see Mazliak (2009).
- 100.
According to certain accounts, including Itô’s notices of the American Mathematical Society ((1998) 45:1455), Lévy’s papers arrived via Romania.
- 101.
Lévy’s American article, in which his stochastic integral is defined, is included in the bibliography of an article by Kakutani (Kakutani, Shinzo (1944) On Brownian motion in n-space. Proc. Imp. Acad. (Tokyo) (November 20)). Lévy’s results apparently arrived in Japan from the USA before Pearl Harbor.
- 102.
Lévy, Paul (1934) Sur les intégrales dont les éléments sont des variables aléatoires indépendantes. Ann. Sc. Norm. Super. Pisa 3:337–366.
- 103.
The notation \(\sqrt{dt}\) had in fact been introduced by Lévy in various publications and appeared in Lévy (1937) where it appears in a note to the chapter entitled Les intégrales à éléments aléatoires indépendants.
- 104.
Number 157–158 of Astérisque (1988).
- 105.
Bernstein, Serge (1934) Principes de la théorie des équations différentielles stochastiques. Труды Физического Математического Института Стеклова 5:95–124 and Bernstein, Serge (1938) Equations différentielles stochastiques. Hermann, Paris.
- 106.
Скороход Анатолий В. (1961) Иследования по теории случайных процессов. Киевский Университет, Киев. English translation (1965): Studies in the theory of random processes. Addison-Wesley.
- 107.
Lévy (1948, pp. 71–72).
- 108.
Lévy, Paul (1941) Intégrales stochastiques. Bull. Soc. Math. Fr.-Sud-Est 67–74.
- 109.
In his seminal article (Itô, Kyoshi (1944) Stochastic integrals. Proc. Imp. Acad. (Tokyo) 20), Itô integrates certain simple processes with respect to Brownian motion. These Riemann sums were the object of an exchange of letters between Itô and Lévy in April 1954. Itô’s construction, especially useful in the space L 2, was later employed for the development of a theory of stochastic integration with respect to martingales. See Dellacherie (1980).
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Barbut, M., Locker, B., Mazliak, L. (2014). Introduction to the Correspondence. In: Paul Lévy and Maurice Fréchet. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, London. https://doi.org/10.1007/978-1-4471-5619-2_1
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DOI: https://doi.org/10.1007/978-1-4471-5619-2_1
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