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 E according to a diffusion equation. We follow the rescaled population of particles, seen as a measure on E.
Scaling limit of a discrete branching process
Simplest setting
For any integer N ≥ 1 {\displaystyle N\geq 1} , consider a branching Brownian process Y N ( t , d x ) {\displaystyle Y^{N}(t,dx)} defined as follows:
Start at t = 0 {\displaystyle t=0} with N {\displaystyle N} independent particles distributed according to a probability distribution μ {\displaystyle \mu } . Each particle independently move according to a Brownian motion. Each particle independently dies with rate N {\displaystyle N} . When a particle dies, with probability 1 / 2 {\displaystyle 1/2} it gives birth to two offspring in the same location. The notation Y N ( t , d x ) {\displaystyle Y^{N}(t,dx)} means should be interpreted as: at each time t {\displaystyle t} , the number of particles in a set A ⊂ R {\displaystyle A\subset \mathbb {R} } is Y N ( t , A ) {\displaystyle Y^{N}(t,A)} . In other words, Y {\displaystyle Y} is a measure-valued random process. Now, define a rescaled process:
X N ( t , d x ) := 1 N Y N ( t , d x ) {\displaystyle X^{N}(t,dx):={\frac {1}{N}}Y^{N}(t,dx)}
Then the finite-dimensional distributions of X N {\displaystyle X^{N}} converge as N → + ∞ {\displaystyle N\to +\infty } to those of a measure-valued random process X ( t , d x ) {\displaystyle X(t,dx)} , which is called a ( ξ , ϕ ) {\displaystyle (\xi ,\phi )} -superprocess, with initial value X ( 0 ) = μ {\displaystyle X(0)=\mu } , where ϕ ( z ) := z 2 2 {\displaystyle \phi (z):={\frac {z^{2}}{2}}} and where ξ {\displaystyle \xi } is a Brownian motion (specifically, ξ = ( Ω , F , F t , ξ t , P x ) {\displaystyle \xi =(\Omega ,{\mathcal {F}},{\mathcal {F}}_{t},\xi _{t},{\textbf {P}}_{x})} where ( Ω , F ) {\displaystyle (\Omega ,{\mathcal {F}})} is a measurable space, ( F t ) t ≥ 0 {\displaystyle ({\mathcal {F}}_{t})_{t\geq 0}} is a filtration, and ξ t {\displaystyle \xi _{t}} under P x {\displaystyle {\textbf {P}}_{x}} has the law of a Brownian motion started at x {\displaystyle x} ). As will be clarified in the next section, ϕ {\displaystyle \phi } encodes an underlying branching mechanism, and ξ {\displaystyle \xi } encodes the motion of the particles. Here, since ξ {\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 Y N ( t , d x ) {\displaystyle Y^{N}(t,dx)} can be much more sophisticated, leading to a variety of superprocesses:
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