Wikipedia / Superprocess
Wikipedia / Superprocess
In probability theory, a superprocess is a measure-valued stochastic process that is usually constructed as a special limit of near-critical branching diffusions.
Informally, a superprocess can be seen as a branching process where each particle splits and dies at infinite rates, and evolves in a state space according to a diffusion equation. We follow the rescaled population of particles, seen as a measure on .
Scaling limit of a discrete branching process
Simplest setting
defined as follows:
.
Each particle independently move according to a Brownian motion.
.
it gives birth to two offspring in the same location.
is a measure-valued random process.
Now, define a rescaled process:
-superprocess, is a measurable space is a filtration).
is a Brownian motion, the resulting object is known as a Super-brownian motion.
Generalization to (ξ, ϕ)-superprocesses
can be much more sophisticated, leading to a variety of superprocesses:
, the state space can now be any Lusin space.
is a càdlàg Markov process (see, Chapter 4, for details).
the associated probability-generating function
-superprocess,.
Commentary on ϕ
, the expected number of offspring is close to 1, and therefore that the process is near-critical.
Generalization to Dawson-Watanabe superprocesses
can be further generalized as follows:
(see, chapter 2, for details).
, it gives birth to offspring according to a measure-valued probability kernel.
which is called a Dawson-Watanabe superprocess,.
Properties
A superprocess has a number of properties. It is a Markov process, and its Markov kernel verifies the branching property is the convolution.A special class of superprocesses are -superprocesses,. A -superprocesses. Its branching mechanism is defined by its factorial moment generating function (the definition of a branching mechanism varies slightly among authors, some in the previous section, others use the factorial moment generating function):
-symmetric stable process with infinitesimal generator.
is a standard Brownian motion-superprocess is called the super-Brownian motion.
One of the most important properties of superprocesses is that they are intimately connected with certain nonlinear partial differential equations When the spatial motion (migration) is a diffusion process, one talks about a superdiffusion. The connection between superdiffusions and nonlinear PDE's is similar to the one between diffusions and linear PDE's.