From 65438 to 0984, after being transferred to Beijing Normal University, Wang Zikun and Li Zhanbing co-chaired the seminar on Markov processes, and continued to engage in research on Markov processes and potential theory, multi-parameter Markov processes and so on. Li Zhanbing1961July graduated from the Department of Mathematical Mechanics of Moscow University, majoring in probability theory and mathematical statistics. 1961August began to teach in the Department of Mathematics of Beijing Normal University. June 1980 to June 1982 visited Massachusetts State University, and June 199 1 visited Kiev University in Ukraine. Li Zhanbing has been engaged in the research of stochastic processes, nonlinear equations and mathematical physics for a long time. In 1980s, the diffusion approximation method he adopted in studying a class of Markov models satisfying a nonlinear Fokker-Planck equation was praised by M.Crandall and R.Gardner. In the study of high-dimensional Burger equation, he once solved a problem put forward by the famous scholar Ya, and he also made many research achievements in the establishment and stability of the master equation of non-equilibrium system, the equation mechanism of basic particles, and the accuracy analysis of cross-location of radiation sources. From 65438 to 0990, Chen Xiong, a doctor trained by Wang Zikun and Li Zhanbing, taught in the Department of Mathematics after graduation, which enriched the research strength of Markov process direction. Chen Xiong's research work mainly focuses on the direction of multi-parameter Markov processes, and has achieved good results in the research of multi-parameter OU processes and multi-parameter Poisson stochastic differential equations. Chen Xiong went abroad to work on 1993, and continued his research in related fields in the following years.
From 65438 to 0988, Wang Zikun received an honorary doctorate in science from Macquarie University in Australia for his achievements in probability theory, science education and research methodology. From the end of 1988 to the beginning of 1989, E.B.Dynkin, an academician of the National Academy of Sciences, was invited to visit China and gave a series of lectures on Dawson-Watanabe superprocess at Nankai University and Beijing Normal University. Since then, Wang Zikun and Li Zhanbing led their research group to start research in this direction. DW superprocess is a mathematical model of random evolution of large-scale microscopic particle population, which has a strong application background in biology, physics and other disciplines. 1989 In a short article, Li Zhanbing expounded Dynkin's conjecture about the integral representation of the branching mechanism of DW superprocess. In 1990, Wang Zikun gave the power series expansion of Laplace functional of DW superprocess. In the same year, Li Zenghu proved the integral representation of branching mechanism, which is one of several basic formulas in the definition of hyperprocess. Dai Jin [Ann]. Maybe. 1993] explains the universality of DW superprocess model with his results: if the superprocess is obtained by taking the limit of a branching particle system, then its branching mechanism must have a specific integral representation. 199 1 year1kloc-0/month, Wang Zikun was elected as an academician of China Academy of Sciences. In the same year, a review of DW superprocesses was published, which systematically introduced the international research progress of measuring Markov processes to domestic academic circles. In this article, Wang Zikun mentioned that the measure branching process with immigrants and multiple groups is a model worth studying. In the paper published in 1992, Li Zenghu introduced a kind of measure bifurcation process with immigration, and studied the convergence of the corresponding particle system. In another paper in the same year, he constructed a general multi-species model by using non-local partial branch DW superprocess. Based on the above ideas, Li Zenghu later cooperated with D.A.Dawson and others to study the construction of non-local partial branch DW super-processes. As applications, he derived a variety of super-processes, such as group type, age structure type, quality structure type and random control type. The unified treatment and concise proof of the existence of the above models are considered as "really good features" by the American Mathematical Review.
In 1993, Wang Zikun defined a class of multi-parameter infinite dimensional OU processes by using multi-parameter random integration, and gave the necessary and sufficient conditions for their absolute continuity. In the same year, Li Zenghu, Li Zhanbing and Wang Zikun gave a complete description of the measured migration process under Feller condition, and obtained the law of large numbers for this kind of process. In 1995, Li Zenghu and Shiga studied the deviation of the branch diffusion process of measure value and the construction of the corresponding migration process. Dawson and Perkins [Mathematics]. Surv。 Monogr。 AMS 1999/Lect。 Notebook math. In 2002, Li Zenghu's and Shiga's theorems about measure value patrol were collected twice to study the "cluster" decomposition of DW superprocesses. Dawson and Gorostiza et al [electron]. J. maybe , 2004] Based on the theoretical framework of migration process given by Li Zenghu and Shiga, the multi-layer group process is deeply studied. In the paper, the author claims that "the existence of immigration process determined by immigration law was originally established by Li and Shiga" and their immigration model "can be incorporated into the framework of immigration process from the boundary of Li and Shiga's research". 1994 10 Li Zenghu stayed in the Department of Mathematics as a postdoctoral fellow after returning from Japan, and has been teaching in the Department of Mathematics since 1996 10.
The usual Dawson-Watanabe superprocess is a mathematical model of random evolution of closed microscopic particle systems. What is more theoretical and practical than this model is the open system model, or the immigration super-process. In 1995, Li Zenghu found that the distribution probability family {n(t):t≥0} of DW superprocesses with migration process satisfies the oblique convolution equation N(r+t) = [N(r)Q(t)]*N(t), where {q (t): t. Li Zeng. As a research tool of open systems, oblique convolution semigroups are also applicable to several other models. For example, Li Zenghu and Dawson later applied oblique convolution semigroups to the study of generalized Mehler semigroups, and gave a complete characterization of Hilbert space-valued ou processes. They also applied the oblique convolution semigroup tool to the study of mathematical finance, and partially answered the regularity problem of affine financial model proposed by Duffie and others. Application probability. 2003], and the relationship between this model and the branching process of random media migration is established. Bojdecki and Gorostiza[ Mathematics. Nachr。 2002] wrote that "Li developed an immigration system theory by introducing and using the concept of oblique convolution semigroups", while Schmuland and Sun [static. Maybe. Lett.200 1] called these results "an important work". Gorostiza called oblique convolution semigroups "play a key role in the process of immigration branching" in German mathematical abstracts.
65438-0996 Beijing Normal University Press reprinted Wang Zikun's "Probability Theory and Application Basis", and his "Random Process Theory, Birth and Death Process and Markov Chain and Brownian Motion and Potential" was edited and reprinted with the title "General Theory of Random Processes". From 65438 to 0999, Wang Zikun studied the absolute continuity and asymptotic behavior of the distribution of multi-parameter infinite dimensional OU processes. His new book Markov Process and Today's Mathematics was published by Hunan Science and Technology Publishing House. 199865438+February to 199965438+February, Li Zenghu visited tokyo institute of technology under the sponsorship of the Postdoctoral Research Fund of the Japan Society for the Promotion of Learning (JSPS), during which he also received additional research funds from the Ministry of Education, Culture, Sports, Science and Technology of Japan, and was invited to visit the Fields Institute of Mathematics in March1999.
From 65438 to 0999, Li Zenghu and Shiga cooperated to give the necessary and sufficient conditions for the reversibility of Fleming-Viot superprocess, thus solving the conjecture put forward by T. Kurtz, an academician of American College of Arts and Sciences. FV superprocess comes from the study of heredity, and the necessary and sufficient conditions for reversibility have always been a long-standing problem in this field. The results of Li Zenghu and others show that the mutation operator of reversible stable distribution genetic system must have some simple form and clear genetic significance. Their research results have triggered the follow-up research on related issues by scholars at home and abroad. For example, Handa [probable]. Theory Related Fields 2002] gives another proof of some results of Li Zenghu et al. under compactness condition, and extends it to the model with recombination. Schmuland and Sun[ Comptes Rendus Mathematical Report of the Royal Canadian Society in 2002] once again proved some of the above results. These authors have repeatedly mentioned in their papers that the problem of reversibility has been solved by Li Zenghu and others, and the necessary and sufficient conditions have been repeatedly proved, which also shows the degree of international concern on this issue.
From 65438 to 0999, Hong and Li Zenghu studied the superprocess of immigration in a random environment, and the nature of the fluctuation limit they discovered and proved was considered as a "surprising" phenomenon by the American Mathematical Review. In 2000, Hong and Wang Zikun studied the law of large numbers and the central limit theorem of the superprocess of branch migration in generalized media. Li Zenghu visited Fields Institute of Mathematics and Carlton University in Canada from June 2000 to May 2006. In 200 1, Li Zenghu and Dawson cooperated to construct a spatial motion-related superprocess by dual method. Hong completed his postdoctoral research at Fudan University in June 2006, returned to China to teach in the Department of Mathematics, and visited Carleton University in Canada in June 2006. In 2002, Wang Zikun won the He Liang Heli Science and Technology Progress Award. Wang Zikun's research in mathematics is mainly in probability theory, and his work is progressing with the development of this subject. He is one of the pioneers of China's probability theory. China's probability theory can have today's international status and has his contribution. Generally speaking, in the early 1960s, he studied the structure of Markov chain, and completely solved the problem of the structure and function distribution of birth and death process. In 1970s, he studied the relationship between Markov process and potential theory, obtained the time and position distribution of Brownian motion and the symmetrical stability process, and studied the statistical prediction of earthquakes. He wrote Brownian Motion and Potential, Probability and Statistical Prediction, etc. In 1980s, he studied multi-index Markov process, introduced the definition of multi-index ornstein-Uhlenbeck process for the first time in the world, and studied its properties. In the early 1990s, in addition to the above work, he also engaged in the research of super-process, which is one of the most active topics in the world at present. The above topics were all important international directions at that time. Keeping up with the development of the times, striving to make achievements in the important frontier of scientific research and striving for the probabilistic significance of achievements and methods are the characteristics of Wang Zikun's mathematical research.
(1) pioneered the probability method of limit transition, and completely solved the construction problem of birth and death process. Random motion starts from 0 and can be extended to infinity, so to determine a random process, it is necessary to observe its motion in infinite time (that is, to give all its finite-dimensional distributions). Can it be determined within a limited time? That is, after observing some so-called "infinitesimal" characteristics of the process in a short time, can these characteristics be used to determine its probability distribution in infinite time? This is the problem to be solved by structuralism. Not every process can be like this. People first began to study some special Markov processes. 1958, almost at the same time, probability theorists W. Feller and Wang Zikan both studied the structure of extinction process, but the methods were different. Ferrer used analytical method, and Wang Zikan used probability method (that is, the limit transition method he pioneered). Therefore, each has its own characteristics. A.a.юшкквич, an expert in probability theory in the Soviet Union, was in the Transaction Fourth Prague Conference on Information Theory, Statistical Decision Functions. Random Processes (1965) commented on page 38 1-387: "Ferrer constructed various extensions of the birth and death process after the orbit reached infinity ... At the same time, Wang Zikun found all extensions of the birth and death process by the method of limit transition." In his discussion with ебд,
(2) 196 1 year, firstly, Wang Zikun studied the distribution of functional and the distribution of stopping time and first arrival time of death process by difference method, and obtained profound results. These two works were later developed by some domestic colleagues and praised by some foreign universities and research institutes. Commenting on the study, Professor D.G.Kendall of Cambridge University said, "I think.
1980, Wang Zikun studied the integral functional by recursive method and published a paper. After the publication of this paper, he received letters from nine countries (United States, France, Federal Republic of Germany, Democratic Republic of Germany, India, Czech Republic, Israel, Netherlands and Italy) 17 unit (university or research institute) asking for a brochure of this paper.
(3) It is used to study the general properties of Markov processes (ergodicity, uniformity, recursion, Martin boundary, etc.). ), see papers [4, 7, 8].
(4) After1980, I studied the relationship between Markov process and potential theory, and published papers [22, 25, 26] and books. 1983, the multi-index Markov process is studied. See references [23, 27 and 30].
(5) In addition to the research on Markov processes, Wang Zikun also initiated the research on stochastic functional analysis in China (see paper [5 1]). Under his leadership, China has done a lot of work in this field.
Most of the research results in (1), (2) and (3) are summarized in Wang Zikun's monograph [4 1, 44].
(6) The multi-index Markov process is studied for the first time in China. The definition of multi-index ornstein-Uhlenbeck process (OUP) was put forward for the first time in the world, and a systematic result was obtained. From single-index process to multi-index process, just as from univariate function to multivariate function, the complexity and difficulty of the problem are greatly increased. OUP is an important stochastic process. It has important applications in physics, but the predecessors only studied the case of single index, and the multi-index OUP was first studied by Wang Zikun. Later, many people continued this research.
Wang Zikun is also engaged in the research of super-process, and has achieved the results of Power Series Expansion of Super-process. In addition, he is deeply interested in "randomness" and "chaos", see papers [35, 39].
(7) There are many kinds of books. Among them, probability theory and its application foundation, stochastic process theory and birth and death process and Markov chain form a complete system from foundation to frontier. The third part is mainly the monograph of Wang Zikun's research results, which is included in the fifth issue of Pure Mathematics and Applied Mathematics by Science Press. For the English revision, see [4 1]. Mathematics Review commented on this book: "This is a beautiful and clear book." These three books have played an important role in promoting the teaching and scientific research of probability theory in China. Some universities (such as Nankai University, Beijing Normal University, Sun Yat-sen University, etc. ) as a teaching material for graduate students, college students and teachers. In this regard, Wang Zikun mainly made the following contributions.
(1) led the academic research of the Statistical Prediction Group of Nankai University, initiated the "random transfer prediction method" and "the method of using foreign major earthquakes to report domestic major earthquake-related areas", and reported some earthquakes many times, which was valued by the State Seismological Bureau and won the second prize of Tianjin Science and Technology. Combined with the earthquake, the pole shift is also studied theoretically (see paper [165438+).
(2) In cooperation with comrades in the army, we have completed the research of simulating stochastic process on computer, put forward a theoretical scheme and compiled a calculation program. Due to relevant regulations, this work has been communicated internally and has not been published publicly.
3. About scientific methods and popular science work.
Wang Zikun believes that teachers should not only impart knowledge, but also cultivate their abilities. Therefore, he attaches great importance to learning methods and research methods, especially the experiences and experiences of famous scholars, which can arouse his interest. 1967+0977, he compiled his speech on learning methods in the 1960s, together with his daily notes, into an article Random Talk on Scientific Discovery, which was published in 1968+0977. 1986, Shanghai People's Publishing House published a separate book, which is a unique reading. Su, an old-timer in the field of mathematics, made an exact evaluation of this book in the preface: "Comrade Wang Zikun surveyed the ancient and modern times, looked at China and foreign countries, selected many meaningful discoveries and facts from the long river of natural science development, tried his best to analyze and summarize them with dialectical materialism and historical materialism, and expounded some basic laws of scientific discovery. And discuss the qualities that a natural science worker should strive to possess. These contents were written by the author in the case of the Gang of Four's metaphysics and idealism, which is particularly commendable. " Su Lao also said, "The author is a mathematician. It is equally valuable to write such works while discussing mathematics."
Vertical and horizontal talk attracts readers with its fresh and unique style, concise and smooth writing style and solid and rich content; Many chapters in the book can be called beautiful and moving prose, with a blend of reason and feeling and endless aftertaste, which makes people intoxicated with the enjoyment of beauty. Some chapters have been selected into middle school Chinese textbooks.
After the lecture, Wang Zikun published dozens of popular science articles in Red Flag magazine, People's Daily, Guangming Daily, China Youth Daily and other newspapers. 1985, he published another book, Boating in Kehai, which also had a great influence on readers.