Superprocess (original) (raw)

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For the opposite of a subprocess in computing, see Parent process.

An ( ξ , d , β ) {\displaystyle (\xi ,d,\beta )} $(\xi ,d,\beta )$ -superprocess, X ( t , d x ) {\displaystyle X(t,dx)} $X(t,dx)$ , within mathematics probability theory is a stochastic process on R × R d {\displaystyle \mathbb {R} \times \mathbb {R} ^{d}} $\mathbb {R} \times \mathbb {R} ^{d}$ 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 {\displaystyle \mathbb {R} } $\mathbb {R}$ .

Scaling limit of a discrete branching process

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Branching Brownian process for N=30

For any integer N ≥ 1 {\displaystyle N\geq 1} $N\geq 1$ , consider a branching Brownian process Y N ( t , d x ) {\displaystyle Y^{N}(t,dx)} $Y^{N}(t,dx)$ defined as follows:

The notation Y N ( t , d x ) {\displaystyle Y^{N}(t,dx)} $Y^{N}(t,dx)$ means should be interpreted as: at each time t {\displaystyle t} $t$ , the number of particles in a set A ⊂ R {\displaystyle A\subset \mathbb {R} } $A\subset \mathbb {R}$ is Y N ( t , A ) {\displaystyle Y^{N}(t,A)} $Y^{N}(t,A)$ . In other words, Y {\displaystyle Y} $Y$ is a measure-valued random process.[1]

Now, define a renormalized 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)} $X^{N}(t,dx):={\frac {1}{N}}Y^{N}(t,dx)$

Then the finite-dimensional distributions of X N {\displaystyle X^{N}} $X^{N}$ converge as N → + ∞ {\displaystyle N\to +\infty } $N\to +\infty$ to those of a measure-valued random process X ( t , d x ) {\displaystyle X(t,dx)} $X(t,dx)$ , which is called a ( ξ , ϕ ) {\displaystyle (\xi ,\phi )} $(\xi ,\phi )$ -superprocess,[1] with initial value X ( 0 ) = μ {\displaystyle X(0)=\mu } $X(0)=\mu$ , where ϕ ( z ) := z 2 2 {\displaystyle \phi (z):={\frac {z^{2}}{2}}} $\phi (z):={\frac {z^{2}}{2}}$ and where ξ {\displaystyle \xi } $\xi$ 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})} $\xi =(\Omega ,{\mathcal {F}},{\mathcal {F}}_{t},\xi _{t},{\textbf {P}}_{x})$ where ( Ω , F ) {\displaystyle (\Omega ,{\mathcal {F}})} $(\Omega ,{\mathcal {F}})$ is a measurable space, ( F t ) t ≥ 0 {\displaystyle ({\mathcal {F}}_{t})_{t\geq 0}} $({\mathcal {F}}_{t})_{t\geq 0}$ is a filtration, and ξ t {\displaystyle \xi _{t}} $\xi _{t}$ under P x {\displaystyle {\textbf {P}}_{x}} ${\textbf {P}}_{x}$ has the law of a Brownian motion started at x {\displaystyle x} $x$ ).

As will be clarified in the next section, ϕ {\displaystyle \phi } $\phi$ encodes an underlying branching mechanism, and ξ {\displaystyle \xi } $\xi$ encodes the motion of the particles. Here, since ξ {\displaystyle \xi } $\xi$ is a Brownian motion, the resulting object is known as a Super-brownian motion.[1]

Generalization to (ξ, ϕ)-superprocesses

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

Instead of R {\displaystyle \mathbb {R} } $\mathbb {R}$ , the state space can now be any Lusin space E {\displaystyle E} $E$ .
The underlying motion of the particles can now be given by ξ = ( Ω , F , F t , ξ t , P x ) {\displaystyle \xi =(\Omega ,{\mathcal {F}},{\mathcal {F}}_{t},\xi _{t},{\textbf {P}}_{x})} $\xi =(\Omega ,{\mathcal {F}},{\mathcal {F}}_{t},\xi _{t},{\textbf {P}}_{x})$ , where ξ t {\displaystyle \xi _{t}} $\xi _{t}$ is a càdlàg Markov process (see,[1] Chapter 4, for details).
A particle dies at rate γ N {\displaystyle \gamma _{N}} $\gamma _{N}$
When a particle dies at time t {\displaystyle t} $t$ , located in ξ t {\displaystyle \xi _{t}} $\xi _{t}$ , it gives birth to a random number of offspring n t , ξ t {\displaystyle n_{t,\xi _{t}}} $n_{t,\xi _{t}}$ . These offspring start to move from ξ t {\displaystyle \xi _{t}} $\xi _{t}$ . We require that the law of n t , x {\displaystyle n_{t,x}} $n_{t,x}$ depends solely on x {\displaystyle x} $x$ , and that all ( n t , x ) t , x {\displaystyle (n_{t,x})_{t,x}} $(n_{t,x})_{t,x}$ are independent. Set p k ( x ) = P [ n t , x = k ] {\displaystyle p_{k}(x)=\mathbb {P} [n_{t,x}=k]} $p_{k}(x)=\mathbb {P} [n_{t,x}=k]$ and define g {\displaystyle g} $g$ the associated probability-generating function: g ( x , z ) := ∑ k = 0 ∞ p k ( x ) z k {\textstyle g(x,z):=\sum \limits _{k=0}^{\infty }p_{k}(x)z^{k}} ${\textstyle g(x,z):=\sum \limits _{k=0}^{\infty }p_{k}(x)z^{k}}$

Add the following requirement that the expected number of offspring is bounded: sup x ∈ E E [ n t , x ] < + ∞ {\displaystyle \sup \limits _{x\in E}\mathbb {E} [n_{t,x}]<+\infty } $\sup \limits _{x\in E}\mathbb {E} [n_{t,x}]<+\infty$ Define X N ( t , d x ) := 1 N Y N ( t , d x ) {\displaystyle X^{N}(t,dx):={\frac {1}{N}}Y^{N}(t,dx)} $X^{N}(t,dx):={\frac {1}{N}}Y^{N}(t,dx)$ as above, and define the following crucial function: ϕ N ( x , z ) := N γ N [ g N ( x , 1 − z N ) − ( 1 − z N ) ] {\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]} $\phi _{N}(x,z):=N\gamma _{N}\left[g_{N}{\Big (}x,1-{\frac {z}{N}}{\Big )}\,-\,{\Big (}1-{\frac {z}{N}}{\Big )}\right]$ Add the requirement, for all a ≥ 0 {\displaystyle a\geq 0} $a\geq 0$ , that ϕ N ( x , z ) {\displaystyle \phi _{N}(x,z)} $\phi _{N}(x,z)$ is Lipschitz continuous with respect to z {\displaystyle z} $z$ uniformly on E × [ 0 , a ] {\displaystyle E\times [0,a]} $E\times [0,a]$ , and that ϕ N {\displaystyle \phi _{N}} $\phi _{N}$ converges to some function ϕ {\displaystyle \phi } $\phi$ as N → + ∞ {\displaystyle N\to +\infty } $N\to +\infty$ uniformly on E × [ 0 , a ] {\displaystyle E\times [0,a]} $E\times [0,a]$ .

Provided all of these conditions, the finite-dimensional distributions of X N ( t ) {\displaystyle X^{N}(t)} $X^{N}(t)$ converge to those of a measure-valued random process X ( t , d x ) {\displaystyle X(t,dx)} $X(t,dx)$ which is called a ( ξ , ϕ ) {\displaystyle (\xi ,\phi )} $(\xi ,\phi )$ -superprocess,[1] with initial value X ( 0 ) = μ {\displaystyle X(0)=\mu } $X(0)=\mu$ .

Provided lim N → + ∞ γ N = + ∞ {\displaystyle \lim _{N\to +\infty }\gamma _{N}=+\infty } $\lim _{N\to +\infty }\gamma _{N}=+\infty$ , that is, the number of branching events becomes infinite, the requirement that ϕ N {\displaystyle \phi _{N}} $\phi _{N}$ converges implies that, taking a Taylor expansion of g N {\displaystyle g_{N}} $g_{N}$ , the expected number of offspring is close to 1, and therefore that the process is near-critical.

Generalization to Dawson-Watanabe superprocesses

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The branching particle system Y N ( t , d x ) {\displaystyle Y^{N}(t,dx)} $Y^{N}(t,dx)$ can be further generalized as follows:

The probability of death in the time interval [ r , t ) {\displaystyle [r,t)} ![{\displaystyle r,t)} of a particle following trajectory ( ξ t ) t ≥ 0 {\displaystyle (\xi _{t})_{t\geq 0}} $(\xi _{t})_{t\geq 0}$ is exp ⁡ { − ∫ r t α N ( ξ s ) K ( d s ) } {\displaystyle \exp \left\{-\int _{r}^{t}\alpha _{N}(\xi _{s})K(ds)\right\}} $\exp \left\{-\int _{r}^{t}\alpha _{N}(\xi _{s})K(ds)\right\}$ where α N {\displaystyle \alpha _{N}} $\alpha _{N}$ is a positive measurable function and K {\displaystyle K} $K$ is a continuous functional of ξ {\displaystyle \xi } $\xi$ (see,[1] chapter 2, for details).
When a particle following trajectory ξ {\displaystyle \xi } $\xi$ dies at time t {\displaystyle t} $t$ , it gives birth to offspring according to a measure-valued probability kernel F N ( ξ t − , d ν ) {\displaystyle F_{N}(\xi _{t-},d\nu )} $F_{N}(\xi _{t-},d\nu )$ . In other words, the offspring are not necessarily born on their parent's location. The number of offspring is given by ν ( 1 ) {\displaystyle \nu (1)} $\nu (1)$ . Assume that sup x ∈ E ∫ ν ( 1 ) F N ( x , d ν ) < + ∞ {\displaystyle \sup \limits _{x\in E}\int \nu (1)F_{N}(x,d\nu )<+\infty } $\sup \limits _{x\in E}\int \nu (1)F_{N}(x,d\nu )<+\infty$ .

Then, under suitable hypotheses, the finite-dimensional distributions of X N ( t ) {\displaystyle X^{N}(t)} $X^{N}(t)$ converge to those of a measure-valued random process X ( t , d x ) {\displaystyle X(t,dx)} $X(t,dx)$ which is called a Dawson-Watanabe superprocess,[1] with initial value X ( 0 ) = μ {\displaystyle X(0)=\mu } $X(0)=\mu$ .

A superprocess has a number of properties. It is a Markov process, and its Markov kernel Q t ( μ , d ν ) {\displaystyle Q_{t}(\mu ,d\nu )} $Q_{t}(\mu ,d\nu )$ verifies the branching property: Q t ( μ + μ ′ , ⋅ ) = Q t ( μ , ⋅ ) ∗ Q t ( μ ′ , ⋅ ) {\displaystyle Q_{t}(\mu +\mu ',\cdot )=Q_{t}(\mu ,\cdot )*Q_{t}(\mu ',\cdot )} $Q_{t}(\mu +\mu ',\cdot )=Q_{t}(\mu ,\cdot )*Q_{t}(\mu ',\cdot )$ where ∗ {\displaystyle *} $*$ is the convolution.A special class of superprocesses are ( α , d , β ) {\displaystyle (\alpha ,d,\beta )} $(\alpha ,d,\beta )$ -superprocesses,[2] with α ∈ ( 0 , 2 ] , d ∈ N , β ∈ ( 0 , 1 ] {\displaystyle \alpha \in (0,2],d\in \mathbb {N} ,\beta \in (0,1]} ![{\displaystyle \alpha \in (0,2],d\in \mathbb {N} ,\beta \in (0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b69bf5997db7c480018b06db4de970d583de3ad1). A ( α , d , β ) {\displaystyle (\alpha ,d,\beta )} $(\alpha ,d,\beta )$ -superprocesses is defined on R d {\displaystyle \mathbb {R} ^{d}} $\mathbb {R} ^{d}$ . Its branching mechanism is defined by its factorial moment generating function (the definition of a branching mechanism varies slightly among authors, some[1] use the definition of ϕ {\displaystyle \phi } $\phi$ in the previous section, others[2] use the factorial moment generating function):

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

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

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

^ a b c d e f g h Li, Zenghu (2011), Li, Zenghu (ed.), "Measure-Valued Branching Processes", Measure-Valued Branching Markov Processes, Berlin, Heidelberg: Springer, pp. 29–56, doi:10.1007/978-3-642-15004-3_2, ISBN 978-3-642-15004-3, retrieved 2022-12-20
^ a b Etheridge, Alison (2000). An introduction to superprocesses. Providence, RI: American Mathematical Society. ISBN 0-8218-2706-5. OCLC 44270365.