Superprocess

An $(\xi,d,\beta)$-superprocess, $X(t,dx)$, within mathematics probability theory is a stochastic process on $\mathbb \times \mathbb^d$ 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 $\mathbb$.

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 $A\subset \mathbb$ is $Y^N(t,A)$. In other words, $Y$ is a measure-valued random process.

Now, define a renormalized process:

$X^N(t,dx):=\fracY^N(t,dx)$

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 $\phi(z):= \frac$ and where $\xi$ is a Brownian motion (specifically, $\xi=(\Omega,\mathcal,\mathcal_t,\xi_t,\textbf_x)$ where $(\Omega,\mathcal)$ is a measurable space, $(\mathcal_t)_$ is a filtration, and $\xi_t$ under $\textbf_x$ 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 $\mathbb$, the state space can now be any Lusin space $E$.
* The underlying motion of the particles can now be given by $\xi=(\Omega,\mathcal,\mathcal_t,\xi_t,\textbf_x)$, 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 $n_$. These offspring start to move from $\xi_t$. We require that the law of $n_$ depends solely on $x$, and that all $(n_)_$ are independent. Set p_k(x)=\mathbb _=k/math> and define $g$ the associated probability-generating function:$g(x,z):=\sum\limits_^\infty p_k(x)z^k$
Add the following requirement that the expected number of offspring is bounded:\sup\limits_\mathbb _+\inftyDefine $X^N(t,dx):=\fracY^N(t,dx)$ as above, and define the following crucial function:\phi_N(x,z):=N\gamma_N \left _N\Big(x,1-\frac\Big)\,-\,\Big(1-\frac\Big)\right/math>Add the requirement, for all $a\geq 0$, that $\phi_N(x,z)$ is Lipschitz continuous with respect to $z$ uniformly on E\times ,a/math>, and that $\phi_N$ converges to some function $\phi$ as $N\to +\infty$ uniformly on E\times ,a/math>.

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 $\lim_\gamma_N = +\infty$, 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 $(\xi_t)_$ is $\exp\left\$ 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 $F_N(\xi_,d\nu)$. 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 $\sup\limits_\int \nu(1)F_N(x,d\nu)<+\infty$.

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 $\alpha\in (0,2],d\in \N,\beta \in (0,1]$. A $(\alpha,d,\beta)$-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):
:$\Phi(s) = \frac(1-s)^+s$
and the spatial motion of individual particles (noted $\xi$ in the previous section) is given by the $\alpha$-symmetric stable process with infinitesimal generator $\Delta_$.

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 $\Delta u-u^2=0\ on\ \mathbb^d.$ 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

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References

Spatial processes

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