Superprocesses over a Stochastic Flow
(Under the direction of Dr. Robert Adler)
Georgios Skoulakis
We study a specific particle system in which particles undergo
random branching and spatial motion. Such systems are best described,
mathematically, via measure valued stochastic processes. As is now quite
standard, we study the "infinite density", or "diffusion" limit of such a
system as both the number of particles in the system and the branching
rate tend to infinity. The limit processes are generally called
"superprocesses". What differentiates our system from the classical
superprocess case, in which the particles move independently of each
other, is that the motions of our particles are affected by the presence
of a global stochastic flow. The sequence of measure valued processes
describing the system has a weak limit, in the setting described above.
This is shown by first proving tightness of the sequence, and then that
all limit points coincide, since they solve a well-posed martingale
problem. The limit of the sequence is called a "flow superprocess".
Using the particle picture formulation of the flow superprocess, we can
study some of its properties. We give formulae for its first two moments
and two macroscopic quantities describing its average behavior. Some
graphs of these quantities for the special case of linear flow are shown.
We also present some simulations in dimensions one and two that indicate
the differences between the flow process and other related measure valued
processes.
4/23/99