An Introduction to Branching Measure-Valued Processes

by E.B. Dynkin

For about half a century, two classes of stochastic processes--Gaussian processes and processes with independent increments--have played an important role in the development of stochastic analysis and its applications. During the last decade, a third class--branching measure-valued (BMV) processes--has also been the subject of much research. A common feature of all three classes is that their finite-dimensional distributions are infinitely divisible, allowing the use of the powerful analytic tool of Laplace (or Fourier) transforms. All three classes, in an infinite-dimensional setting, provide means for study of physical systems with infinitely many degrees of freedom. This is the first monograph devoted to the theory of BMV processes. Dynkin first constructs a large class of BMV processes, called superprocesses, by passing to the limit from branching particle systems. Then he proves that, under certain restrictions, a general BMV process is a superprocess. A special chapter is devoted to the connections between superprocesses and a class of nonlinear partial differential equations recently discovered by Dynkin.

134 pages, Hardcover

First published January 1, 1994

About the author

Eugene Borisovich Dynkin (Russian: Евгений Борисович Дынкин4) is a Soviet and American mathematician. He has made contributions to the fields of probability and algebra, especially semisimple Lie groups, Lie algebras, and Markov processes. The Dynkin diagram, the Dynkin system, and Dynkin's lemma are named for him.