# Superprocess

An $(\xi ,d,\beta )$-superprocess, $X(t,dx)$, within mathematics probability theory is a stochastic process on $\mathbb {R} \times \mathbb {R} ^{d}$ 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:
* Start at $t=0$ with $N$ independent particles distributed according to a probability distribution $\mu$.
* Each particle independently move according to a Brownian motion.
* Each particle independently dies with rate $N$.
* When a particle dies, with probability $1/2$ it gives birth to two offspring in the same location.
The notation $Y^{N}(t,dx)$ means should be interpreted as: at each time $t$, the number of particles in a set $A\subset \mathbb {R}$ is $Y^{N}(t,A)$. In other words, $Y$ is a measure-valued random process.
Now, define a renormalized process:
$X^{N}(t,dx):={\frac {1}{N}}Y^{N}(t,dx)$
Then the finite-dimensional distributions of $X^{N}$ converge as $N\to +\infty$ 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 $\phi (z):={\frac {z^{2}}{2}}$ and where $\xi$ is a Brownian motion (specifically, $\xi =(\Omega ,{\mathcal {F}},{\mathcal {F}}_{t},\xi _{t},{\textbf {P}}_{x})$ where $(\Omega ,{\mathcal {F}})$ is a measurable space, $({\mathcal {F}}_{t})_{t\geq 0}$ is a filtration, and $\xi _{t}$ under ${\textbf {P}}_{x}$ 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:
* Instead of $\mathbb {R}$, the state space can now be any Lusin space $E$.
* The underlying motion of the particles can now be given by $\xi =(\Omega ,{\mathcal {F}},{\mathcal {F}}_{t},\xi _{t},{\textbf {P}}_{x})$, where $\xi _{t}$ is a càdlàg Markov process (see, Chapter 4, for details).
* A particle dies at rate $\gamma _{N}$
* When a particle dies at time $t$, located in $\xi _{t}$, it gives birth to a random number of offspring $n_{t,\xi _{t}}$. These offspring start to move from $\xi _{t}$. We require that the law of $n_{t,x}$ depends solely on $x$, and that all $(n_{t,x})_{t,x}$ are independent. Set $p_{k}(x)=\mathbb {P} [n_{t,x}=k]$ and define $g$ the associated probability-generating function: g(x, z):=∑$_{k=0}$$^∞$p$_k$(x)z$^k${\textstyle g(x, z):=\sum \limits _{k=0}^{\infty }p_{k}(x)z^{k}}
Add the following requirement that the expected number of offspring is bounded:$\sup \limits _{x\in E}\mathbb {E} [n_{t,x}]<+\infty$Define $X^{N}(t,dx):={\frac {1}{N}}Y^{N}(t,dx)$ as above, and define the following crucial function:$\phi _{N}(x,z):=N\gamma _{N}\left[g_{N}{\Bigx,1-{\frac {z}{N}}{\Big )}\,-\,{\Big1-{\frac {z}{N}}{\Big )}\right]$Add the requirement, for all $a\geq 0$, that $\phi _{N}(x,z)$ is Lipschitz continuous with respect to $z$ uniformly on $E\times [0,a]$, and that $\phi _{N}$ converges to some function $\phi$ as $N\to +\infty$ uniformly on $E\times [0,a]$.
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 $\lim _{N\to +\infty }\gamma _{N}=+\infty$, 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:
* The probability of death in the time interval $[r,t)$ of a particle following trajectory $(\xi _{t})_{t\geq 0}$ is $\exp \left\{-\int _{r}^{t}\alpha _{N}(\xi _{s})K(ds)\right\}$ where $\alpha _{N}$ is a positive measurable function and $K$ is a continuous functional of $\xi$ (see, chapter 2, for details).
* When a particle following trajectory $\xi$ dies at time $t$, it gives birth to offspring according to a measure-valued probability kernel $F_{N}(\xi _{t-},d\nu )$. In other words, the offspring are not necessarily born on their parent's location. The number of offspring is given by $\nu (1)$. Assume that $\sup \limits _{x\in E}\int \nu (1)F_{N}(x,d\nu )<+\infty$.
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 $Q_{t}(\mu ,d\nu )$ verifies the branching property:$Q_{t}(\mu +\mu ',\cdot )=Q_{t}(\mu ,\cdot )*Q_{t}(\mu ',\cdot )$where $*$ is the convolution.A special class of superprocesses are $(\alpha ,d,\beta )$-superprocesses, with $\alpha \in (0,2],d\in \mathbb {N} ,\beta \in (0,1]$. A $(\alpha ,d,\beta )$-superprocesses is defined on $\mathbb {R} ^{d}$. 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):
    $\Phi (s)={\frac {1}{1+\beta }}(1-s)^{1+\beta }+s$
and the spatial motion of individual particles (noted $\xi$ in the previous section) is given by the $\alpha$-symmetric stable process with infinitesimal generator $\Delta _{\alpha }$.
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 $\Delta u-u^{2}=0\ on\ \mathbb {R} ^{d}.$ 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

* Eugene B. Dynkin (2004). Superdiffusions and positive solutions of nonlinear partial differential equations. Appendix A by J.-F. Le Gall and Appendix B by I. E. Verbitsky. University Lecture Series, 34. American Mathematical Society. ISBN 9780821836828.