An -superprocess, X(t,dx), 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 \mathbb{R}.

Scaling limit of a discrete branching process

Simplest setting

For any integer N\geq 1, consider a branching Brownian process Y^N(t,dx) defined as follows: The notation Y^N(t,dx) means should be interpreted as: at each time t, the number of particles in a set is Y^N(t,A). In other words, Y is a measure-valued random process. Now, define a renormalized process: Then the finite-dimensional distributions of X^N converge as to those of a measure-valued random process X(t,dx), which is called a (\xi,\phi)-superprocess, with initial value X(0) = \mu, where and where \xi is a Brownian motion (specifically, where is a measurable space, is a filtration, and \xi_t under has the law of a Brownian motion started at x). As will be clarified in the next section, \phi encodes an underlying branching mechanism, and \xi encodes the motion of the particles. Here, since \xi is a Brownian motion, the resulting object is known as a Super-brownian motion.

Generalization to (ξ, ϕ)-superprocesses

Our discrete branching system Y^N(t,dx) can be much more sophisticated, leading to a variety of superprocesses: Add the following requirement that the expected number of offspring is bounded:Define as above, and define the following crucial function:Add the requirement, for all a\geq 0, that \phi_N(x,z) is Lipschitz continuous with respect to z uniformly on, and that \phi_N converges to some function \phi as uniformly on. Provided all of these conditions, the finite-dimensional distributions of X^N(t) converge to those of a measure-valued random process X(t,dx) which is called a (\xi,\phi)-superprocess, with initial value X(0) = \mu.

Commentary on ϕ

Provided, that is, the number of branching events becomes infinite, the requirement that \phi_N converges implies that, taking a Taylor expansion of g_N, the expected number of offspring is close to 1, and therefore that the process is near-critical.

Generalization to Dawson-Watanabe superprocesses

The branching particle system Y^N(t,dx) can be further generalized as follows: Then, under suitable hypotheses, the finite-dimensional distributions of X^N(t) converge to those of a measure-valued random process X(t,dx) which is called a Dawson-Watanabe superprocess, with initial value X(0) = \mu.

Properties

A superprocess has a number of properties. It is a Markov process, and its Markov kernel verifies the branching property:where * is the convolution.A special class of superprocesses are -superprocesses, with. A -superprocesses is defined on \R^d. Its branching mechanism is defined by its factorial moment generating function (the definition of a branching mechanism varies slightly among authors, some use the definition of \phi in the previous section, others use the factorial moment generating function): and the spatial motion of individual particles (noted \xi in the previous section) is given by the \alpha-symmetric stable process with infinitesimal generator. The \alpha = 2 case means \xi is a standard Brownian motion and the (2,d,1)-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.

Further resources