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An -superprocess, , within mathematics probability theory is a stochastic process on that is usually constructed as a special limit of near-critical branching diffusions.
Informally, it can be seen as a branching process where each particle splits and dies at infinite rates, and evolves according to a diffusion equation, and we follow the rescaled population of particles, seen as a measure on .
For any integer , consider a branching Brownian process defined as follows:
The notation means should be interpreted as: at each time , the number of particles in a set is . In other words, is a measure-valued random process. [1]
Now, define a renormalized process:
Then the finite-dimensional distributions of converge as to those of a measure-valued random process , which is called a -superprocess, [1] with initial value , where and where is a Brownian motion (specifically, where is a measurable space, is a filtration, and under has the law of a Brownian motion started at ).
As will be clarified in the next section, encodes an underlying branching mechanism, and encodes the motion of the particles. Here, since is a Brownian motion, the resulting object is known as a Super-brownian motion. [1]
Our discrete branching system can be much more sophisticated, leading to a variety of superprocesses:
Add the following requirement that the expected number of offspring is bounded:
Provided all of these conditions, the finite-dimensional distributions of converge to those of a measure-valued random process which is called a -superprocess, [1] with initial value .
Provided , that is, the number of branching events becomes infinite, the requirement that converges implies that, taking a Taylor expansion of , the expected number of offspring is close to 1, and therefore that the process is near-critical.
The branching particle system can be further generalized as follows:
Then, under suitable hypotheses, the finite-dimensional distributions of converge to those of a measure-valued random process which is called a Dawson-Watanabe superprocess, [1] with initial value .
A superprocess has a number of properties. It is a Markov process, and its Markov kernel verifies the branching property:
and the spatial motion of individual particles (noted in the previous section) is given by the -symmetric stable process with infinitesimal generator .
The case means is a standard Brownian motion and the -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. The simplest such equation is 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.
This article has multiple issues. Please help
improve it or discuss these issues on the
talk page. (
Learn how and when to remove these template messages)
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An -superprocess, , within mathematics probability theory is a stochastic process on that is usually constructed as a special limit of near-critical branching diffusions.
Informally, it can be seen as a branching process where each particle splits and dies at infinite rates, and evolves according to a diffusion equation, and we follow the rescaled population of particles, seen as a measure on .
For any integer , consider a branching Brownian process defined as follows:
The notation means should be interpreted as: at each time , the number of particles in a set is . In other words, is a measure-valued random process. [1]
Now, define a renormalized process:
Then the finite-dimensional distributions of converge as to those of a measure-valued random process , which is called a -superprocess, [1] with initial value , where and where is a Brownian motion (specifically, where is a measurable space, is a filtration, and under has the law of a Brownian motion started at ).
As will be clarified in the next section, encodes an underlying branching mechanism, and encodes the motion of the particles. Here, since is a Brownian motion, the resulting object is known as a Super-brownian motion. [1]
Our discrete branching system can be much more sophisticated, leading to a variety of superprocesses:
Add the following requirement that the expected number of offspring is bounded:
Provided all of these conditions, the finite-dimensional distributions of converge to those of a measure-valued random process which is called a -superprocess, [1] with initial value .
Provided , that is, the number of branching events becomes infinite, the requirement that converges implies that, taking a Taylor expansion of , the expected number of offspring is close to 1, and therefore that the process is near-critical.
The branching particle system can be further generalized as follows:
Then, under suitable hypotheses, the finite-dimensional distributions of converge to those of a measure-valued random process which is called a Dawson-Watanabe superprocess, [1] with initial value .
A superprocess has a number of properties. It is a Markov process, and its Markov kernel verifies the branching property:
and the spatial motion of individual particles (noted in the previous section) is given by the -symmetric stable process with infinitesimal generator .
The case means is a standard Brownian motion and the -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. The simplest such equation is 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.