Defining a probability space with a poisson point process

Question

The issue is formally defining some objects

Consider a fixed (at the moment) value $p \in [0,1]$.

We've a poisson point process in $\mathbb{R}^d$ with rate $1$, we shall call it $\eta$. We'd like to assign to every point $x \in \eta$ a random value $\omega(x) \in \{black, white\}$ such that $$P(\omega(x) = black)=p \; P(\omega(x) = white) = 1-p.$$

My question is: How do I define this probability space? Afterwards I'd like to put $\omega(x)\sim U[0,1]$ in order to couple models with different parameters, so how would I define it?

One way to define formally a poisson point process is considering it a random measure, that is, a random variable taking values in the set of measures in $\mathbb{R}^d$ (with some properties to make it "poisson"). In this case I need a random something else that gives me the number of points in a region $A \subseteq \mathbb{R}^d$ and it's value (either black, white or the uniform variable).

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Context to the question:

I'm reading an article about Voronoi percolation and I've some problem understanding how I'll be able do make a coupling in this case.

There are two ways this process is defined: Fix a parameter $p \in [0,1]$:

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1- Consider a poisson point process $\eta$ in $\mathbb{R}^d$ with rate $1$, I'll call this points nodes. Define for each node $x \in \eta$ a voronoi cell $$C(x):=\left\{y \in \mathbb{R}^d: \|x-y\| = \min_{x' \in \eta}\|x'-y\|\right\}.$$

making a Voronoi Tesselation of $\mathbb{R}^d$.

After that we assign to each node $x \in \eta$ a random colour $\omega(x) \in \{black, white\}$ such that $P(\omega(x)= black) = p$ and $P(\omega(x)=white) = 1-p$.

Finally color every cell according to the color of it's node. That is, if $y \in C(x)$ then $\omega(y) = \omega(x)$.

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2- Consider 2 poisson processes in $\mathbb{R}^d$ with parameters $p$ and $1-p$. We'll call them $\eta^b$ and $\eta^w$, denoting the black and white nodes respectivelly. Take the voronoi tesselation made by $\eta = \eta^b \cup \eta^w$ and color the cells in the same way, according to the nodes.

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Both definitions are "equivalent" but there's a catch. In the second definition the process $\eta$ depends on $p$ inherently, whilst in the first definition the coloring happens afterwards, making it possible to consider different colorings (with different parameter p) with the same nodes (which is indeed needed in the article).

So the question is: What is the measurable space in definition 1?

Answer 1

The object you are looking for is named marked Poisson process.

In fact, Poisson point process can be defined in rather absract spaces (from now I'll rather use the term Poisson random measure).

I'll be a bit informal but I recommand you to take a look to the book of Andreas Kypaprianou on Lévy processes which has a nice introduction on general Poisson random measure. The idea is that a Poisson random measure $\mathcal{N}$ on a measurable space $E$ with intensity measure $\mu$ (which is a measure on $E$) is a random variable with value in the space of measure on $E$ (so a random measure) which satifies

• For two disjoint measurable set $A$ and $B$, $\mathcal{N}(A)$ and $\mathcal{N}(B)$ are independent.
• The law of $\mathcal{N}(A)$ is Poissonian with parameter $\mu(A)$.

In your situation, you would consider a Poisson random measure on the space $E=\mathbb{R}^{d}\times\{black, white\}$ with intensity measure given by $Leb\otimes (p\delta_{black}+(1-p)\delta_{white})$.

So, for instance, what is the probability that exactly one black point stands in the unit open ball of $\mathbb{R}^d$ ? It's $\mathbb{P}(\mathcal{N}(B(0,1)\times\{black\})=1)$ but $\mathcal{N}(B(0,1)\times\{black\})$ is Poissonian with parameter $Leb\otimes (p\delta_{black}+(1-p)\delta_{white})\left(B(0,1)\times\{black\}\right)=|B(0,1)|\times p.$

You can also take a look to the books : Introduction to the theory of point processes vol.1 and vol.2 (Daley, D.J., and Vere-Jones, D.)