A short sketch of life and research of A.V. Skorokhod

Anatolii Volodymyrovych Skorokhod was born on September, 10th, 1930 in Nikopol, an industrial city of Dnipropetrovs'k region on the south of Ukraine. His father, Volodymyr Oleksiyovych, was a teacher of mathematics, physics and astronomy. His mother, Nadiya Andriivna, besides mathematics, taught also history, literature, music. According to Nadiya Andriivna's recollections, their children (they had two sons) lived in the atmosphere of the various interests of their parents, with love to books and nature. The parents treated them with care, with respect to their desires and inclinations. Probably, it is not by chance that the smaller son, Valeriї, who admired his elder brother, also chose the scientific career for himself and became an academician in physics.

The parents worked mostly in different villages and small miner's towns, from time to time the family moved to a new place. In 1935 they settled in the city of Marganets. Here, in 1937, Anatoliї went to school. Schooling was interrupted by the war, and Anatoliї had to continue his studies at home.

In 1946, the family temporarily moved to Kovel (Volyn region of Western part of Ukraine) fleeing from the famine in the Dnipro region. Instead of boundless steppe burnt by sun with water-melon fields and corn plantations known from the childhood, there were forests and coppices. The contrast in the tenor of life was striking. Here, in Volyn, folk traditions were kept, polyphonic choruses in which very young singers side by side with the adults carefully performed their parts were especially impressing. Although Anatoliї lived here, in Kovel, for quite a short time, the nature of this region and especially the national spirit of the people made great impression on him. At that time, he imagined his future as a long voyage captain, but the medical panel found his near sight, so he had to renounce this romantic dream.

In 1948, Anatolii graduated from a secondary school (with a golden medal) in Kovel and entered Taras Shevchenko Kyiv State University, Department of Mechanics and Mathematics.

Studying was easy for him, it was interesting. His aptitude for research first manifested itself when he was a student. Skorokhod decided to specialize in probability theory at the Department of Mathematical Analysis. These investigations were carried out under the substantial influence of Prof. B.V.Gnedenko and Prof. I.I.Gikhman (who later became a close friend and colleague of Skorokhod). The young man actively joined the scientific work, he managed to think over solution of several problems at the same time. On graduating from the University, Skorokhod became the author of five scientific papers, two of them were published in the leading scientific journals ``Успехи матем. наук'' [``Soviet Math. Surveys''] and `` Докл. Акад. наук СССР'' [``Soviet Math. Dokl.''], the rest two were published in the collection of scientific works of the students of Kyiv University. It is worth to mention that two of these early Skorokhod's works were translated into English and published in Selected Translations on Mathematical Statistics and Probability (1961).

On graduating from the University, Skorokhod went to the post-graduate studies, to Moscow to take postgraduate courses under the guidance of Prof. E.B. Dynkin at the Moscow University (1953-56). It was a period of swift development of investigations in the field of the probability theory at Moscow University when foundations were laid for the theory of random processes. A large group of talented young people organized around

A.N.Kolmogorov. Among them, the personality of a young scientist from Kyiv was distinguished by his profound knowledge and a lot of unexpected ideas. They say that young colleagues tried to use every possibility (for example, in a long queue in the University cafeteria) to contact Skorokhod and usually each of them got an answer to his question on some problem.

Skorokhod's works of this period were abounded in original approaches for the problem solving and unusual associations. It was at that time when he proposed the topology in a space of functions without discontinuities of the second order. This topology served as an instrument for proving limit theorems for the wide class of random processes, now it has the name Skorokhod topology in the world literature. He created a principally new approach to the proof of the limit theorems (the method of a single probability space that became well known too. This is a direct probability method of investigation, its objects are random variables but not their distribution functions, as it was before. The characteristic feature of Skorokhod's research was his urge to find the final result, the necessary and sufficient conditions of the statements.

In 1957, Skorokhod returned to Kyiv and began his work as a lecturer at the Kyiv University. In 1964, he became the Head of the Department of the Theory of Random Processes at the Institute of Mathematics of the Ukrainian Academy of Sciences, continuing as a lecturer at the Kyiv University. For his scientific results, Skorokhod received numerous titles and degrees: Doctor of Sciences, Professor (1963), Corresponding Member of the Ukrainian Academy of Sciences (1967), Academician of the Ukrainian Academy of Sciences (1985), and a Member of the American Academy of Art and Science (2000). In 1982 and in 2003, he was awarded the Ukrainian State Prize in Science and Technology.

From the very beginning of his work at the Kyiv University, Skorokhod distinguished himself by a unique manner of delivering lectures; proving numerous statements impromptu, he made his students the participants of creative scientific work. When Skorokhod returned to Kyiv, the work f the scientific seminar on probability theory at the Kyiv University became much more active. His discussions with speakers and capability to understand the core of a problem, generalize it, find possible weak points in the proof, and reveal the hidden relation of the problem considered to other problems turned the seminar sessions into a real creative laboratory, and all interested scholars tried to deliver a talk at the seminar in the presence of Skorokhod. Thus, the Kyiv school in probability theory was largely formed as a result of Skorokhod's activities.

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Since the middle 1950s, Skorokhod's works have played a fundamental role in development of the theory of random processes; to a great extent, they determined the directions of further investigations in this theory not only in the Ukraine but also in the entire world.

The first series of Skorokhod's works that gained him wide recognition was devoted to the limit theorems for random processes constructed on the basis of sums of independent random variables. These works accomplished the series of attempts of numerous mathematicians aimed at the generalization of the famous Donsker invariance principle to the case where the limit process is an arbitrary, not necessarily continuous, process with independent increments. In these works, Skorokhod demonstrated his extraordinary power of independent thought and constructive imagination. In these papers, which formed the basis for his Candidate-Degree thesis, Skorokhod proposed the method of a single probability space (mentioned earlier) and introduced several topologies in the space of functions that do not have discontinuities of the second kind, one of which is now widely known as the Skorokhod topology. These tools enabled him to completely solve all problems related to the aforementioned generalization of the Donsker invariance principle.

As early as in the works indicated, Skorokhod demonstrated his inclination in favor of direct probability methods for solving problems of probability theory. In the preface to his first monograph ``Studies in the Theory of Random Processes'' (Kyiv University, Kyiv, 1961), he wrote that the problem of choice of a particular group of methods makes sense only with respect to an individual problem; the advantage of analytic methods lies in their universality, whereas the advantage of probability methods is their close relation to the essence of the problem.

The theory of stochastic differential equations is the most significant branch of probability theory in which direct probability methods are largely used, and it is quite natural that this theory drew Skorokhod's attention. As a result, he immediately obtained several significant results, which made him one of the leading experts in this branch of mathematics. Among these results, one should mention his proof of the theorem on existence of solutions of stochastic differential equations by the method of a single probability space under the assumption that the coefficients of these equations are continuous functions (i.e., they may not satisfy the Lipschitz condition).

Another important direction in the theory of stochastic differential equations in which Skorokhod obtained fundamental pioneer results at the beginning of the 1960s is related to the equations that describe processes in manifolds with boundary. These results aroused much interest all over the world and stimulated numerous deep investigations on the problem of construction of processes of this type. Later, Skorokhod continued the investigation of this problem [see his monograph ``Stochastic Equations for Complex Systems'' (Nauka, Moscow, 1983)]. In this monograph, Skorokhod also considered another problem that had drawn his attention in the second half of the 1960s, namely, the problem of description of the local structure of all continuous Markov processes or, say, processes that do not have discontinuities of the second kind. In 1966, he proved that a sufficiently broad class of continuous Markov processes can be reduced to quasidiffusion processes by a random change of the time variable. In the monograph ``Stochastic Equations for Complex Systems'', he constructed stochastic differential equations for quasidiffusion processes taking values in spaces of complex structure (e.g., manifolds with boundary, manifolds with variable dimensionality, etc.).

Among the works of Skorokhod published in the 1970s, one should mention the books ``Integration in Hilbert Spaces'', ``Random Linear Operators'', and ``Theory of Random Processes'' (in 3 volumes; written together with I.I.Gikhman)  a fundamental monograph reflecting the contemporary state of the most branches of the theory of random processes.

In the 1970s, Skorokhod introduced several notions, which are now widely used not only by mathematicians, but also by physicists. Among them, one should mention the notions of extended stochastic integral (the Skorokhod integral), strong (weak) random linear operator, and stochastic semigroup. The notion of strong random linear operator was used by Skorokhod for the description of the structure of certain classes of stochastic semigroups. These results were published in the monographs ``Processes with Independent Increments'' (2nd edition, Nauka, Moscow, 1986) and ``Asymptotic Methods of the Theory of Stochastic Differential Equations'' (Naukova Dumka, Kyiv, 1987). In the latter monograph, Skorokhod applied the notion of stochastic semigroup to the problem of stability of stochastic systems.

Skorokhod's contribution to the formation of the Ukrainian school in probability theory can hardly be overestimated. He has more than 50 disciples, among which there are 17 Doctors of Sciences. His lectures on all branches of contemporary theory of random processes presented at the Kyiv University and numerous popular-science works contributed much to the mathematical education of youth. Skorokhod is the author of 23 scientific monographs (and, there are 22 translations of these monographs) and more than 300 works published in scientific journals, he headed numerous scientific seminars, etc.

Skorokhod paid considerable attention to widespread mathematical knowledge. He wrote textbooks and popular-science books (as a whole, he is an author of 16 textbooks and popular-science books), he delivered lectures on television for students. Every September, he lectured schoolchildren at the opening ceremony of a new school-year of the University for Young Mathematicians which worked at the Institute of Mathematics in the seventies and eighties. With deep understanding, he supported popularizing the names of distinguished Ukrainian mathematicians of the past. For this goal, he undertook several travels to lecture all over Ukraine. This supported creation of museums or other memorial places in honour of outstanding mathematicians of the past (for example, G. Voronoї, V. Bunyakovskiї, M. Kravchuk).

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Skorokhod always was distinguished by his independent opinion. He always stood his ground, though this was quite dangerous under the totalitarian regime. In 1968, he took part in the action of the group of Ukrainian intellectuals defending the constitutional rights of citizens of the country. All the participants of this action were censured. As a result, Skorokhod was not allowed to lecture students, advise post-graduates, he was excluded from the Editorial Boards of some scientific journals, and for fifteen years, he was not permitted to participate in scientific conferences abroad. Skorokhod stood this forced limitation of his own rights with proper pride. He told himself that time, that mathematics saved him from all the life troubles. During those fifteen years of disfavour, he worked particularly fruitfully. His absence at international scientific conferences gave birth to the opinion among foreign scientists that ``Skorokhod'' was the collective name of a group of Soviet scientists, just as the group of French mathematicians united under the name ``Bourbaki''.

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Since 1993, Skorokhod has worked at the Michigan State University (Lansing, Michigan, USA), retaining close scientific relations with the Institute of Mathematics of the Ukrainian Academy of Sciences. His scientific works of recent years are devoted to the investigation of the asymptotic behavior of dynamical systems under random perturbations.

The main strength of Anatolii Volodymyrovych Skorokhod as a scientist is his thorough thinking over every day, and his unceasing quest for new mathematical truth. Owing to his intense work day after day, the creative spark given to him from God is now a bright shining star of the first magnitude on the mathematical frontier.