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Lecture 18: Particle Processes

Introduction

Recall from last time:

A set process is a point process on $(\mathcal{F}', \mathcal{B}(\mathcal{F}'))$, the space of nonempty closed subsets of $\mathbb{R}^d$.
A set process $ = {n} {F_n} $ is stationary if $ t_x := {n} {F_n - x}, , x ^d. $

Today:

Poisson set processes
Particle processes
Particle processes as marked point processes

From now on, all set processes are assumed simple with locally finite intensity measure.

Poisson Set Processes

Let $\Lambda$ be a locally finite measure on $\mathcal{F}'$: ($ (_C) < , C. $)

Then there exists a Poisson set process $\Phi$ with intensity measure $\Lambda$.

The capacity functional of the coverage model $ Z_ $ is $ T_{Z_}(C) = 1 - ((_C)=0) = 1 - e^{-(_C)}. $

Proposition. A Poisson set process is stationary iff its intensity measure is translation invariant: \[ \Lambda(A) = \Lambda(\{F+x : F\in A\}), \qquad \forall A\in\mathcal{B}(\mathcal{F}'),\; x\in\mathbb{R}^d. \]

Particle Processes

Definition (Particle Process). A particle process $\Phi$ in $\mathbb{R}^d$ is a (simple) point process in $\mathcal{F}'$ with support in $\mathcal{C}'$ (nonempty compact sets).

For stationary particle processes, there is a useful decomposition of the intensity measure.

For $C\in\mathcal{C}'$,
Let $c(C)$ denote the center of the smallest closed ball containing $C$.

Then, $ ’_0 := { C’ : c(C)=0 }. $ is called the grain space.

Define the mapping \[ T : \mathbb{R}^d \times \mathcal{C}'_0 \to \mathcal{C}', \qquad T(x,C) = x + C. \]

Theorem. Let $\Phi$ be a stationary particle process on $\mathbb{R}^d$ with locally finite, nonzero intensity measure $\Lambda$.
Then there exist a number $\lambda \in (0,\infty)$ and a probability measure $\mathbb{Q}$ on $\mathcal{C}'_0$ such that \[ \Lambda = \lambda T\big( \ell \otimes Q \big), \] where $\ell$ denotes the Lebesgue measure on $\mathbb{R}^d$.

In particular, \[ \int_{\mathcal{C}'} f \, d\Lambda = \lambda \int_{\mathcal{C}'_0} \int_{\mathbb{R}^d} f(C+x)\, dx \, \mathbb{Q}(dC). \]

$\lambda$ is called the intensity of $\Phi$.
$\mathbb{Q}$ is called the grain distribution of $\Phi$.

In general, one may consider any center function $c : \mathcal{C}' \to \mathbb{R}^d$ such that:

$c$ is measurable,
$c$ is compatible with translations: \[ c(C+x) = c(C) + x, \qquad \forall\, x \in \mathbb{R}^d. \]

One still obtains a decomposition of $\Lambda$, but with different grain distributions.

Definition. A random closed set with distribution $\mathbb{Q}$ is called a typical grain of $\Phi$.

Particle Processes as Marked Point Processes

Theorem. If $\Phi = \sum_{n=1}^{\infty} \delta_{Z_n}$ is a stationary particle process with locally finite intensity measure $\Lambda$, and $c$ is a center function, then:

$\Phi_0 := \sum_{n=1}^{\infty} \delta_{c(Z_n)}$ is a stationary point process on $\mathbb{R}^d$;
$\widetilde{\Phi} := \sum_{n=1}^{\infty} \delta_{(c(Z_n),\, Z_n - c(Z_n))}$ is a stationary point process on $\mathbb{R}^d$ with mark space $\mathcal{C}'_0$
(i.e., a stationary point process on $\mathbb{R}^d \times \mathcal{C}'_0$).

Remarks: Let $\mathbb{Q}$ be the grain distribution of $\Phi$ and let $\lambda$ be its intensity.

We can interpret $\mathbb{Q}$ as the distribution of the mark of the point at the origin under the Palm distribution of $\Phi_0$.
Campbell’s theorem implies: for any $A \in \mathcal{B}(\mathcal{C}')$, \[ \mathbb{Q}(A) = \frac{1}{\lambda} \mathbb{E}\!\left[ \sum_{n=1}^{\infty} \mathbf{1}_{\{ c(Z_n) \in [0,1]^d \}} \mathbf{1}_{\{ Z_n - c(Z_n) \in A \}} \right]. \]
In particular, \[ \lambda = \mathbb{E}\!\left[ \sum_{n=1}^{\infty} \mathbf{1}_{\{ c(Z_n) \in [0,1]^d \}} \right], \] and $\lambda$ is also the intensity of $\Phi_0$.

Germ-Grain Processes

Definition. A marked point process $\widetilde{\Phi}$ with mark space $\mathcal{C}'$ is called a germ–grain process if \[ \Phi := \sum_{(x,C)\in \mathrm{supp}(\widetilde{\Phi})} \delta_{x+C} \] is a particle process. We need to check local finiteness.

Proposition. If $\widetilde{\Phi}$ is a stationary marked point process with mark space $\mathcal{C}'$ and mark distribution $\mathbb{Q}$, then the particle process $\Phi$ defined above has locally finite intensity measure $\Lambda$ if and only if \[ \int_{\mathcal{C}'} \mathrm{Vol}(C + B^d)\, \mathbb{Q}(dC) < \infty, \] where $B^d$ denotes the ball of radius $1$ in $\mathbb{R}^d$ centered at $0$, and
\[ C + B^d = \{ x + y : x \in C,\ y \in B^d \} \] is the Minkowski sum.

If a germ–grain process $\widetilde{\Phi}$ is obtained by $\mathbb{Q}$-marking of an independent point process $\Phi_0$ on $\mathbb{R}^d$, then
$\widetilde{\Phi}$ is called an independent germ–grain process.

Boolean Model

Let \[ \widetilde{\Phi} = \sum_{n\in\mathbb{N}} \delta_{(X_n, Z_n)} \] be a stationary, independent germ–grain process such that the ground point process \[ \Phi_0 = \sum_{n\in\mathbb{N}} \delta_{X_n} \] is a stationary Poisson process.

Note:
This implies that $\widetilde{\Phi}$ is a Poisson marked point process.

The coverage model given by \[ \widetilde{Z} := \bigcup_{n\in\mathbb{N}} (Z_n + X_n) \] is called a Boolean model.

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