Simplest setting

Branching Brownian process for N=30

For any integer ${\displaystyle N\geq 1}$, consider a branching Brownian process ${\displaystyle Y^{N}(t,dx)}$ defined as follows:

• Start at ${\displaystyle t=0}$ with ${\displaystyle N}$ independent particles distributed according to a probability distribution ${\displaystyle \mu }$.
• Each particle independently move according to a Brownian motion.
• Each particle independently dies with rate ${\displaystyle N}$.
• When a particle dies, with probability ${\displaystyle 1/2}$ it gives birth to two offspring in the same location.

The notation ${\displaystyle Y^{N}(t,dx)}$ means should be interpreted as: at each time ${\displaystyle t}$, the number of particles in a set ${\displaystyle A\subset \mathbb {R} }$ is ${\displaystyle Y^{N}(t,A)}$. In other words, ${\displaystyle Y}$ is a measure-valued random process.^[1]

Now, define a renormalized process:

${\displaystyle X^{N}(t,dx):={\frac {1}{N}}Y^{N}(t,dx)}$

Then the finite-dimensional distributions of ${\displaystyle X^{N}}$ converge as ${\displaystyle N\to +\infty }$ to those of a measure-valued random process ${\displaystyle X(t,dx)}$, which is called a ${\displaystyle (\xi ,\phi )}$-superprocess,^[1] with initial value ${\displaystyle X(0)=\mu }$, where ${\displaystyle \phi (z):={\frac {z^{2}}{2}}}$ and where ${\displaystyle \xi }$ is a Brownian motion (specifically, ${\displaystyle \xi =(\Omega ,{\mathcal {F}},{\mathcal {F}}_{t},\xi _{t},{\textbf {P}}_{x})}$ where ${\displaystyle (\Omega ,{\mathcal {F}})}$ is a measurable space, ${\displaystyle ({\mathcal {F}}_{t})_{t\geq 0}}$ is a filtration, and ${\displaystyle \xi _{t}}$ under ${\displaystyle {\textbf {P}}_{x}}$ has the law of a Brownian motion started at ${\displaystyle x}$).

As will be clarified in the next section, ${\displaystyle \phi }$ encodes an underlying branching mechanism, and ${\displaystyle \xi }$ encodes the motion of the particles. Here, since ${\displaystyle \xi }$ is a Brownian motion, the resulting object is known as a Super-brownian motion.^[1]

Generalization to (ξ, ϕ)-superprocesses

Our discrete branching system ${\displaystyle Y^{N}(t,dx)}$ can be much more sophisticated, leading to a variety of superprocesses:

• Instead of ${\displaystyle \mathbb {R} }$, the state space can now be any Lusin space ${\displaystyle E}$.
• The underlying motion of the particles can now be given by ${\displaystyle \xi =(\Omega ,{\mathcal {F}},{\mathcal {F}}_{t},\xi _{t},{\textbf {P}}_{x})}$, where ${\displaystyle \xi _{t}}$ is a càdlàg Markov process (see,^[1] Chapter 4, for details).
• A particle dies at rate ${\displaystyle \gamma _{N}}$
• When a particle dies at time ${\displaystyle t}$, located in ${\displaystyle \xi _{t}}$, it gives birth to a random number of offspring ${\displaystyle n_{t,\xi _{t}}}$. These offspring start to move from ${\displaystyle \xi _{t}}$. We require that the law of ${\displaystyle n_{t,x}}$ depends solely on ${\displaystyle x}$, and that all ${\displaystyle (n_{t,x})_{t,x}}$ are independent. Set ${\displaystyle p_{k}(x)=\mathbb {P} [n_{t,x}=k]}$ and define ${\displaystyle g}$ the associated probability-generating function:${\textstyle g(x,z):=\sum \limits _{k=0}^{\infty }p_{k}(x)z^{k}}$

Add the following requirement that the expected number of offspring is bounded:

$${\displaystyle \sup \limits _{x\in E}\mathbb {E} [n_{t,x}]<+\infty }$$

Define ${\displaystyle X^{N}(t,dx):={\frac {1}{N}}Y^{N}(t,dx)}$ as above, and define the following crucial function:

$${\displaystyle \phi _{N}(x,z):=N\gamma _{N}\left[g_{N}{\Big (}x,1-{\frac {z}{N}}{\Big )}\,-\,{\Big (}1-{\frac {z}{N}}{\Big )}\right]}$$

Add the requirement, for all ${\displaystyle a\geq 0}$, that ${\displaystyle \phi _{N}(x,z)}$ is Lipschitz continuous with respect to ${\displaystyle z}$ uniformly on ${\displaystyle E\times [0,a]}$, and that ${\displaystyle \phi _{N}}$ converges to some function ${\displaystyle \phi }$ as ${\displaystyle N\to +\infty }$ uniformly on ${\displaystyle E\times [0,a]}$.

Provided all of these conditions, the finite-dimensional distributions of ${\displaystyle X^{N}(t)}$ converge to those of a measure-valued random process ${\displaystyle X(t,dx)}$ which is called a ${\displaystyle (\xi ,\phi )}$-superprocess,^[1] with initial value ${\displaystyle X(0)=\mu }$.

Generalization to Dawson-Watanabe superprocesses

The branching particle system ${\displaystyle Y^{N}(t,dx)}$ can be further generalized as follows:

• The probability of death in the time interval ${\displaystyle [r,t)}$ of a particle following trajectory ${\displaystyle (\xi _{t})_{t\geq 0}}$ is ${\displaystyle \exp \left\{-\int _{r}^{t}\alpha _{N}(\xi _{s})K(ds)\right\}}$ where ${\displaystyle \alpha _{N}}$ is a positive measurable function and ${\displaystyle K}$ is a continuous functional of ${\displaystyle \xi }$ (see,^[1] chapter 2, for details).
• When a particle following trajectory ${\displaystyle \xi }$ dies at time ${\displaystyle t}$, it gives birth to offspring according to a measure-valued probability kernel ${\displaystyle F_{N}(\xi _{t-},d\nu )}$. In other words, the offspring are not necessarily born on their parent's location. The number of offspring is given by ${\displaystyle \nu (1)}$. Assume that ${\displaystyle \sup \limits _{x\in E}\int \nu (1)F_{N}(x,d\nu )<+\infty }$.

Then, under suitable hypotheses, the finite-dimensional distributions of ${\displaystyle X^{N}(t)}$ converge to those of a measure-valued random process ${\displaystyle X(t,dx)}$ which is called a Dawson-Watanabe superprocess,^[1] with initial value ${\displaystyle X(0)=\mu }$.