Outline for today:
Notation:
Definition (\(k\)-Flat Processes). A \(k\)-flat process in \(\mathbb{R}^d\) is a simple point process on \(\mathcal{F}'\) with (locally finite) intensity measure concentrated on \(\mathcal{A}(d,k)\).
As with particle processes, stationary \(k\)-flat processes exhibit a decomposition of their intensity measure.
Theorem. Let \(\Phi\) be a stationary \(k\)-flat process in \(\mathbb{R}^d\) with locally finite intensity measure \(\ell \neq 0\) on \(\mathcal{A}(d,k)\).
Then there exists a number \(\gamma \in (0,\infty)\) and a probability measure \(\mathbb{Q}\) on \(\mathcal{G}(d,k)\) such that for all measurable \(f : \mathcal{A}(d,k) \to \mathbb{R}_{+}\), \[
\int_{\mathcal{A}(d,k)} f\,d\ell
=
\gamma
\int_{\mathcal{G}(d,k)}
\int_{L^{\perp}}
f(L+x)\,\ell_{L^{\perp}}(dx)\,
\mathbb{Q}(dL).
\]
Let \(\pi_{0} : \bigcup_{k=1}^{d-1} \mathcal{A}(d,k) \to \bigcup_{k=1}^{d-1} \mathcal{G}(d,k)\) map any affine subspace to its translate through the origin.
Then, for any \(A \in \mathcal{B}(\mathcal{G}(d,k))\), \[ \gamma\,\mathbb{Q}(A) = \frac{1}{\kappa_{d-k}} \mathbb{E}\!\left[ \Phi\!\left( \mathcal{F}_{B^{d}} \cap \pi_0^{-1}(A) \right) \right]. \]
In particular, \[ \gamma = \frac{1}{\kappa_{d-k}} \mathbb{E}\!\left[ \Phi\!\left( \mathcal{F}_{B^{d}} \right) \right], \] and \[ \mathbb{Q}(A) = \frac{ \mathbb{E}\!\left[ \Phi\!\left( \mathcal{F}_{B^{d}}\cap \pi_0^{-1}(A) \right) \right] }{ \mathbb{E}\!\left[ \Phi\!\left( \mathcal{F}_{B^{d}} \right) \right] }. \] This highlights why \(\mathbb{Q}\) is called the directional distribution.
Proposition. Let \(\Phi\) be a stationary \(k\)-flat process in \(\mathbb{R}^d\) with intensity \(\gamma\).
Then, for all \(B \in \mathcal{B}(\mathbb{R}^d)\), \[
\mathbb{E}\!\left[
\sum_{E \in \Phi} \ell_{E}(B)
\right]
=
\gamma\,\ell(B).
\]
Proof. By Campbell’s theorem and the decomposition of the intensity measure \(\Lambda\), \[ \mathbb{E}\!\left[ \sum_{E \in \Phi} \ell_{E}(B) \right] = \int_{\mathcal{A}(d,k)} \ell_{E}(B)\,\Lambda(dE). \]
{(d,k)} {L^{}} {L+x}(B),{L^{}}(dx), (dL) ]
_{(d,k)} (B),(dL) = ,(B). ]
\(\square\)