Eugene B. Dynkin
Ph.D. (1948), Dr. of Science (1951) Moscow University
Probability theory, Lie groups
Lie groups were the main subject of my earlier research. Dynkin's Diagrams are widely used by mathematicians and physicists. After 1954, probability theory became the central field of my interests. Principal efforts were devoted to Markov processes and their connections with potential theory and partial differential equations. Other work includes research in mathematical statistics (sufficient statistics, exponential families), optimal control (optimal stopping, control with incomplete data) and mathematical economics (economic growth and economic equilibrium under uncertainty).
In the 80s I worked on the relationship between Markov processes and random fields that arise in statistical physics and quantum field theory. One of the results — an isomorphism theorem connecting Gaussian fields with local times for Markov processes — has a considerable impact on the work of a number of investigators. For the last decade, my main efforts are devoted to the theory of measure-valued branching processes. (The name superprocesses suggested by me for these processes is now standard in mathematical literature.) Connections between superdiffusions and a class of nonlinear partial differential equations were established that makes it possible to apply powerful analytic tools for investigating the path behavior of superdiffusions, and that provides a new probabilistic approach to problems of nonlinear PDEs. New directions — the description of all positive solutions of a certain class of nonlinear equations and the study of removable boundary singularities of such solutions — have been started in a series of joint papers of Dynkin and Kuznetsov. A theory developed by them and by a number of other investigators is presented in a systematic way in a monograph of Dynkin published in 2002.
The complete classification of positive solutions of nonlinear equations $\Deltau=u^\alpha$ with $1<\alpha\le 2$ in a bounded smooth domain resulted from a series of papers by Dynkin and by Dynkin and Kuznetsov written in 2003. A systematic presentation of these results is contained in a book of Dynkin published in 2004.
An Introduction to Branching Measure-Valued Processes, CRM Monograph Series 6, American Mathematical Society, Providence, RI, 1994.
Fine topology and fine trace on the boundary associated with a class of semilinear differential equations (with S. E. Kuznetsov), Comm. Pure and Appl. Math. 51 (1998), 897–936.
Selected Papers of E. B. Dynkin with Commentary, Amer. Math. Soc. and International Press, 2000.
Diffusions, Superdiffusions and Partial Differential Equations, AMS Colloquium Publications, Vol. 50, Providence, Rhode Island, 2002.
Superdiffusions and positive solutions of nonlinear partial differential equations, American Mathematical Society, University Lecture Series, Volume 34, 2004.
An application of probability to nonlinear analysis, Proceedings of the Second Abel Symposium, Oslo 2005 (to appear).