Superprocess

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 ${\displaystyle \mathbb {R} }$.

Simplest setting

 

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.

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, 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.

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, 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:

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

 

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

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

 

${\displaystyle a\geq 0}$ 

${\displaystyle \phi _{N}(x,z)}$ 

${\displaystyle z}$ 

${\displaystyle E\times [0,a]}$ 

${\displaystyle \phi _{N}}$ 

${\displaystyle \phi }$ 

${\displaystyle N\to +\infty }$ 

${\displaystyle E\times [0,a]}$ 

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, with initial value ${\displaystyle X(0)=\mu }$ .

Commentary on ϕ

Provided ${\displaystyle \lim _{N\to +\infty }\gamma _{N}=+\infty }$ , that is, the number of branching events becomes infinite, the requirement that ${\displaystyle \phi _{N}}$  converges implies that, taking a Taylor expansion of ${\displaystyle g_{N}}$ , 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 ${\displaystyle Y^{N}(t,dx)}$  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, 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, with initial value ${\displaystyle X(0)=\mu }$ .

A superprocess has a number of properties. It is a Markov process, and its Markov kernel ${\displaystyle Q_{t}(\mu ,d\nu )}$  verifies the branching property:

$${\displaystyle Q_{t}(\mu +\mu ',\cdot )=Q_{t}(\mu ,\cdot )*Q_{t}(\mu ',\cdot )}$$

 

${\displaystyle *}$ 

${\displaystyle (\alpha ,d,\beta )}$ -superprocesses

${\displaystyle \alpha \in (0,2],d\in \mathbb {N} ,\beta \in (0,1]}$ 

${\displaystyle (\alpha ,d,\beta )}$ -superprocesses

${\displaystyle \mathbb {R} ^{d}}$ 

${\displaystyle \phi }$ 

${\displaystyle \Phi (s)={\frac {1}{1+\beta }}(1-s)^{1+\beta }+s}$ 

and the spatial motion of individual particles (noted ${\displaystyle \xi }$  in the previous section) is given by the ${\displaystyle \alpha }$ -symmetric stable process with infinitesimal generator ${\displaystyle \Delta _{\alpha }}$ .

The ${\displaystyle \alpha =2}$  case means ${\displaystyle \xi }$  is a standard Brownian motion and the ${\displaystyle (2,d,1)}$ -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 ${\displaystyle \Delta u-u^{2}=0\ on\ \mathbb {R} ^{d}.}$  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.

• 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.