Superprocess

An [math]\displaystyle{ (\xi,d,\beta) }[/math]-superprocess, [math]\displaystyle{ X(t,dx) }[/math], within mathematics probability theory is a stochastic process on [math]\displaystyle{ \mathbb{R} \times \mathbb{R}^d }[/math] 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 [math]\displaystyle{ \mathbb{R} }[/math].

Scaling limit of a discrete branching process

Simplest setting

Branching Brownian process for N=30

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

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

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

Now, define a renormalized process:

[math]\displaystyle{ X^N(t,dx):=\frac{1}{N}Y^N(t,dx) }[/math]

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

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

Generalization to (ξ, ϕ)-superprocesses

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

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

Add the following requirement that the expected number of offspring is bounded:[math]\displaystyle{ \sup\limits_{x\in E}\mathbb{E}[n_{t,x}]\lt +\infty }[/math]Define [math]\displaystyle{ X^N(t,dx):=\frac{1}{N}Y^N(t,dx) }[/math] as above, and define the following crucial function:[math]\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] }[/math]Add the requirement, for all [math]\displaystyle{ a\geq 0 }[/math], that [math]\displaystyle{ \phi_N(x,z) }[/math] is Lipschitz continuous with respect to [math]\displaystyle{ z }[/math] uniformly on [math]\displaystyle{ E\times [0,a] }[/math], and that [math]\displaystyle{ \phi_N }[/math] converges to some function [math]\displaystyle{ \phi }[/math] as [math]\displaystyle{ N\to +\infty }[/math] uniformly on [math]\displaystyle{ E\times [0,a] }[/math].

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

Commentary on ϕ

Provided [math]\displaystyle{ \lim_{N\to+\infty}\gamma_N = +\infty }[/math], that is, the number of branching events becomes infinite, the requirement that [math]\displaystyle{ \phi_N }[/math] converges implies that, taking a Taylor expansion of [math]\displaystyle{ g_N }[/math], 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 [math]\displaystyle{ Y^N(t,dx) }[/math] can be further generalized as follows:

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

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

Properties

A superprocess has a number of properties. It is a Markov process, and its Markov kernel [math]\displaystyle{ Q_t(\mu,d\nu) }[/math] verifies the branching property:[math]\displaystyle{ Q_t(\mu+\mu',\cdot) = Q_t(\mu,\cdot)*Q_t(\mu',\cdot) }[/math]where [math]\displaystyle{ * }[/math] is the convolution.A special class of superprocesses are [math]\displaystyle{ (\alpha,d,\beta) }[/math]-superprocesses, with [math]\displaystyle{ \alpha\in (0,2],d\in \N,\beta \in (0,1] }[/math]. A [math]\displaystyle{ (\alpha,d,\beta) }[/math]-superprocesses is defined on [math]\displaystyle{ \R^d }[/math]. 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 [math]\displaystyle{ \phi }[/math] in the previous section, others use the factorial moment generating function):

[math]\displaystyle{ \Phi(s) = \frac{1}{1+\beta}(1-s)^{1+\beta}+s }[/math]

and the spatial motion of individual particles (noted [math]\displaystyle{ \xi }[/math] in the previous section) is given by the [math]\displaystyle{ \alpha }[/math]-symmetric stable process with infinitesimal generator [math]\displaystyle{ \Delta_{\alpha} }[/math].

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