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 $$\mathbb{R}$$.

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:

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:$$\phi_N(x,z):=N\gamma_N \left[g_N\Big(x,1-\frac{z}{N}\Big)\,-\,\Big(1-\frac{z}{N}\Big)\right]$$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 $$E\times [0,a]$$.
 * 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:$g(x,z):=\sum\limits_{k=0}^\infty p_k(x)z^k$

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 $$ \alpha\in (0,2],d\in \N,\beta \in (0,1]$$. 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):
 * $$ \Phi(s) = \frac{1}{1+\beta}(1-s)^{1+\beta}+s$$

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.