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.
Readership
Research mathematicians and graduate students.