Z Z Z
1{d(Ty−x+z+tbϕ, x)> t}Γ(ψ\δy, dϕ)fp(x−z−tb−y) g(x−z−tb−y, ψ)ψ(dy)µ(dψ)
= Z Z Z
1{d(ϕ, y+z+tb)> t}Γ(ψ\δy, dϕ)fp(x−z−tb−y) g(x−z−tb−y, ψ)ψ(dy)µ(dψ).
Assuming that fp and g are bounded and continuous and that µ has finite intensity measure, we see that the last expression converges to
Z Z Z
1{d(ϕ, y+z)>0}Γ(ψ\δy, dϕ)fp(x−z−y)g(x−z−y, ψ)ψ(dy)µ(dψ) as t→+0. This has been announced in Remark 4.19.
5 Stationary grain models
In this section we assume that the grain model Ξ isstationary, i.e. that the distribution of Ξ +x is the same for allx∈Rd. We find it convenient (see [24]) to express stationarity in terms of an abstract measurable flowθ :Rd×Ω→Ω for which the bijectionsθx : Ω→Ω, x∈Rd, defined by θx(·) :=θ(x,·) satisfy the flow propertyθx◦θy =θx+y for allx, y ∈Rd. We then assume that the probability measure P is invariantunder all θx and that
Ξ(ω)−x= Ξ(θxω), x∈Rd.
The invariance property (2.3), which is also enjoyed by the additive extensions of the support measures, implies that the random measures Cj+(Ξ,·), j = 0, . . . , d−1, inherit stationarity from Ξ, i.e.
Cj+(Ξ(ω),(A+x)×C) =Cj+(Ξ(θxω), A×C), (5.1)
for all measurable A, C ⊂Rd. In particular it follows that the intensity measures Λ+j are of product form if Λ+j(· ×Rd) is locally finite. Assuming that the intensity
λ+j :=E
Cj+(Ξ,[0,1]d×Rd) is finite, we have
Λ+j(d(x, b)) = λ+j Hd(dx)Rj(db), where Rj is a probability measure on Rd. In fact, we find that
Rj(·) = (λ+j)−1Λ+j ([0,1]d× ·)
ifλ+j >0. Note thatR :=Rd−1is the rose of directions introduced in the previous section.
Due to stationarity, the volume fraction ¯p := P(x ∈ Ξ) and the contact distribution function
HB(t, A) :=P(d(x)≤t, u(x)∈A|x /∈Ξ)
are independent of x for all measurable A ⊂ Rd. Therefore, in the present stationary situation Corollary 4.6 implies that
∂
∂t t=+0
(1−p)H¯ B(t, A) = 2λs,+d−1 Z
1{∇hBˇ(u)∈A}hBˇ(u)Rsd−1(du).
Our next aim is to introduce stochastic kernels κj, j = 0, . . . , d−1, from Ω×Rd to Rd satisfying
Cj+(Ξ, d(z, b)) =κj(z, db)Cj+(Ξ, dz×Rd) P −a.s. (5.2) and being stationary in the sense that
κj(ω, z,·) = κj(θzω,0,·), (ω, z)∈Ω×Rd. (5.3) To reach that goal we take a measurable setD⊂Rd with positive and finite volume. By (5.1) and the assumption of invariance,
Qj(·) := 1 Hd(D)
Z Z
1{z ∈D}1{(θzω, b)∈ ·}Cj+(Ξ(ω), d(z, b))P(dω) (5.4) defines a measure on Ω×Rdwhich is independent ofD. In the terminology of Mecke [23], Qj(· ×A) is the Palm measure of the stationary random measure Cj+(Ξ,· ×A), where A⊂Rd is measurable. If λ+j >0, then
Pj(·) := (λ+j )−1Qj(· ×Rd)
is thePalm probabilityofCj+(Ξ,·×Rd). Just for completeness we definePj :=P ifλ+j = 0.
As in [34] we might interpret Pj as a conditional probability measure given that 0 is a typical point of Cj+(Ξ,· ×Rd). Equation (5.4) implies the refined Campbell theorem
Z Z
1{(ω, b, x)∈ ·}Qj(d(ω, b))Hd(dx) = Z Z
1{(θzω, b, z)∈ ·}Cj+(Ξ(ω), d(z, b))P(dω).
(5.5)
Next we introduce a stochastic kernel ˜κj from Ω toRd by disintegrating Qj according to Qj(d(ω, b)) = ˜κj(ω, db)Qj(dω×Rd).
The definition κj(ω, z,·) := ˜κj(θzω,·), (ω, z)∈Ω×Rd, yields a kernel with the property (5.3) while equation (5.2) follows from a straightforward application of the refined Campell theorem (5.5).
The next result expresses expectations with respect to the stationary probability mea-sure P in terms of the Palm probabilities Pj.
Theorem 5.1 For any measurable f : Ω→[0,∞),
where Ej denotes expectation with respect to the Palm probability Pj.
Proof. Denote byW an arbitrary subset of Rd with Hd(W) = 1. Stationarity gives E[1{d(0)>0}f] =E that the right-hand side of equation (5.6) is equal to
d−1 and the refined Campbell theorem for Pj we obtain that the last sum equals
d−1
which can be simplified to yield the right hand-side of the asserted equality.
Theorem 5.1 allows several remarks and corollaries (cf. [19]). Using the probability measures
Proof. Apply Theorem 5.1 with f :=1{(d(0), u(0))∈(0, r]×A}.
The corollary says that (1−p)H¯ B(·, A) is absolutely continuous with density t 7→
d−1
X
j=0
(d−j)bd−jλ+j td−j−1Gj((t,∞]×A). (5.7) The fact that the contact distributionHB(·,Rd) is absolutely continuous has been proved in [10] using Federer’s coarea theorem.
Remark 5.3 It follows directly from the definitions that Ej[κj(0,·)] =Gj((0,∞]× ·) =Rj(·)
provided that λ+j > 0, since Gj({0} ×Rd) = 0. Hence the value of the density (5.7) for t= 0 equals 2λ+d−1R(A), which is in accordance with Theorem 4.1.
In the remainder of this section we discuss our results in terms of the marked point process Φ =P∞
n=1δ(ξn,Zn) and the point process Ψ = Φ(· × Kd) introduced in the previ-ous section. A result by Weil and Wieacker [38] justifies that we can assume that Φ is stationary, i.e. we assume that
∞
X
n=1
δ(ξn◦θx,Zn◦θx) =
∞
X
n=1
δ(ξn−x,Zn)
holds for all x∈Rd. In other words, (5.1) is satisfied withCj+(Ξ(·),·) replaced by Φ. We also assume that the intensity
λΨ:=E
Ψ([0,1]d)
is strictly positive and finite. The intensity measure α of Φ then satisfies α(d(x, K)) =λΨHd(dx)Q0(dK),
where Q0 is a probability measure on Kd which is called the distribution of the typical grain.
Just for simplicity we assume that Ψ is simple, i.e. Ψ({x}) ≤ 1 for all x ∈ Rd. We can then define a Kd-valued stochastic process {Z(x) : x ∈Rd} by letting Z(x) :=Zn if x =ξn for some n ∈ N and Z(x) := K0 otherwise, where K0 is some fixed convex body.
This process is stationary in the sense that Z(x) = Z(0)◦θx, x ∈ Rd. Denoting by PΨ the Palm probability of the point process Ψ, we have the following version of the refined Campbell theorem:
λΨ Z Z
1{(ω, x, Z(0))∈ ·}PΨ(dω)Hd(dx) = Z Z
1{(θxω, x, Z(x))∈ ·}Ψ(ω, dx)P(dω).
(5.8) LetK 7→PΨK be a version of the conditional probability PΨ(·|Z(0) =K). By (5.8),
P(x,K)=PΨK◦θx, (x, K)∈Rd× Kd, (5.9) is one possible choice of the Palm probabilities introduced in the previous section, where we note that Q0 = PΨ(Z(0) ∈ ·). From Theorem 4.16 we easily obtain the following result.
Proposition 5.4 Let the assumptions of Proposition 4.9 be satisfied and let A ⊂ Rd be measurable. Then (1−p)H¯ B(·, A) is absolutely continuous with density
t7→
Remark 5.5 If the second-order reduced moment measure R
(ψ \ δ0)(·)PΨ(Ψ ∈ dψ) is absolutely continuous and Φ is an independent marking of Ψ, then the second factorial moment measure α(2) of Φ satisfies the assumption of Proposition 4.9.
Example 5.6 In Propositions 4.13 (see also Remark 4.14) and 4.26 we found explicit expressions for the direction dependent contact distribution of the inhomogeneous Boolean model. Under the additional assumption of stationarity, the functionf which appears in these propositions is equal to the constant λΨ. We write
V¯j := for the mean of the total jth Minkowski support measure and
S¯j(·) :=
Z
Cj(K,Rd× ·)Q0(dK) for the mean jth Minkowski surface area measure. Then we obtain
HB(t) :=HB(t,Rd) = 1−exp
Of course, the latter equation can also be derived from Proposition 5.4 using Slivnyak’s theorem PΨ(Ψ\δ0 ∈ ·) = P(Ψ ∈ ·) and taking into account that Φ also under PΨ is an independent marking of Ψ.
In the course of the proof of Proposition 4.13 we have especially shown that Z
Hd(K+rBd)Q0(dK)<∞, r >0,
is a consequence of our general assumption (II). Therefore we can assume that Z
Vj(K)Q0(dK)<∞, j ∈ {0, . . . , d−1},
where Vj(·), j ∈ {0, . . . , d}, are the classical (Euclidean) quermass-integrals.
But then holds true for an arbitrary convex body B containing the origin (see also [41]). For example, if B is a line segment, then we obtain the linear contact distribution. These extensions can be established by approximating B from outside by a decreasing sequence of strictly convex bodies. But, of course, this can also be verified more directly.
If, in addition, Q0 is invariant with respect to rotations, then one can use (5.3.24) from [28] as well as (independent of x∈Rd) to deduce the well-known relation
ρB(t) =λΨ
This completes the discussion of the homogeneous Boolean model.
Finally, we generalize Proposition 5.4. The assertion is similar to Theorem 5.1 and follows by combining the methods of the proofs of Theorem 5.1 and Theorem 4.16. The details are left to the reader.
Theorem 5.7 Let the assumptions of Proposition 4.9 be satisfied. For any measurable f : Ω→[0,∞), Acknowledgement. This work has been initiated during a very stimulating conference on stochastic geometry in Oberwolfach. We would like to thank Professor Rolf Schneider for helpful discussions on the subjects of this paper and in particular for suggesting the second assertion of Theorem 3.4. We also thank Professor Dietrich Stoyan for drawing our attention to the subject of Theorem 4.1.
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Authors’ addresses:
Daniel Hug, Mathematisches Institut, Albert-Ludwigs-Universit¨at, Eckerstr. 1, D-79104 Freiburg i. Br., Germany, e-mail: hug@sun1.mathematik.uni-freiburg.de
G¨unter Last, Institut f¨ur Mathematische Stochastik, Technische Universit¨at Braunschweig, Pockelsstr. 14, Postfach 3329, D-38106 Braunschweig, Germany, e-mail: g.last@tu-bs.de