Superprocess

In probability theory, a superprocess is a measure-valued stochastic process that is usually constructed as a special limit of near-critical branching diffusions. Informally, a superprocess can be seen as a branching process where each particle splits and dies at infinite rates, and evolves in a state space according to a diffusion equation. We follow the rescaled population of particles, seen as a measure on .

In probability theory, a superprocess is a measure-valued stochastic process that is usually constructed as a special limit of near-critical branching diffusions. Informally, a superprocess can be seen as a branching process where each particle splits and dies at infinite rates, and evolves in a state space according to a diffusion equation. We follow the rescaled population of particles, seen as a measure on . 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 rescaled 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. 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 Add the following requirement that the expected number of offspring is bounded: \sup\limits_{x\in E}\mathbb{E}[n_{t,x}] 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] . 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 . 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. 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 . 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. ==Further resources==

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