ANATOLII SKOROKHOD
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 watermelon 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 postgraduate studies, to
Moscow to take postgraduate courses under the guidance of Prof.
E.B. Dynkin at the Moscow University (195356). 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.
* * * * * * *
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 CandidateDegree 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 popularscience 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 popularscience books (as a whole, he
is an author of 16 textbooks and popularscience books), he delivered
lectures on television for students. Every September, he lectured schoolchildren
at the opening ceremony of a new schoolyear 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).
* * * * *
* *
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 postgraduates, 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''.
* * * * *
* *
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.
