Superprocess Explained

An

(\xi,d,\beta) -superprocess, X(t,dx) , within mathematics probability theory is a stochastic process on R x 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

R .

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 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): = (1)/(N)} Y ^{N(t,dx)}

Then the finite-dimensional distributions of

X ^{N} converge as N\to +inf ty 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): = (z ^{2})/(2) and where \xi is a Brownian motion (specifically, \xi = (\Omega, l{F}, l{F} _{t,\xi} _{t,bf{P}} _{x)} where (\Omega, l{F}) is a measurable space, ( l{F} _{t)} _{t\geq} is a filtration, and \xi _{t} under bf{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:

• Instead of

R , the state space can now be any Lusin space E .

• The underlying motion of the particles can now be given by

\xi = (\Omega, l{F}, l{F} _{t,\xi} _{t,bf{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)=P[n} _{t,x} = k] and define g the associated probability-generating function:$g(x,z):=\backslash sum\backslash limits\_^\backslash infty\; p\_k(x)z^k$Add the following requirement that the expected number of offspring is bounded:Define X ^{N(t,dx): = (1)/(N)} Y ^{N(t,dx)} as above, and define the following crucial function:$$\backslash phi\_N(x,z):=N\backslash gamma\_N\; \backslash left[g\_N\backslash Big(x,1-\backslash frac\{z\}\{N\}\backslash Big)\backslash ,-\backslash ,\backslash Big(1-\backslash frac\{z\}\{N\}\backslash Big)\backslash right]$$Add the requirement, for all a\geq 0 , that \phi _{N(x,z)} is Lipschitz continuous with respect to z uniformly on E x [ 0 ,a] , and that \phi _{N} converges to some function \phi as N\to +inf ty uniformly on E x [ 0 ,a] .

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} = +inf ty , 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} is \exp\left\{ - \int ^{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} \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

Q _{t(\mu,d\nu)} verifies the branching property:$$Q\_t(\backslash mu+\backslash mu\text{'},\backslash cdot)\; =\; Q\_t(\backslash mu,\backslash cdot)*Q\_t(\backslash mu\text{'},\backslash cdot)$$where * is the convolution.A special class of superprocesses are (\alpha,d,\beta) -superprocesses,^[1] 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) = (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} 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.

Further resources

• Book: 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 . 9780821836828.

References

1. Book: Etheridge, Alison . An introduction to superprocesses . 2000 . American Mathematical Society . 0-8218-2706-5 . Providence, RI . 44270365.