Z Z Z

1{d(T_{y−x+z+tb}ϕ, x)> t}Γ(ψ\δ_{y}, dϕ)f_{p}(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\δ_{y}, dϕ)f_{p}(x−z−tb−y)
g(x−z−tb−y, ψ)ψ(dy)µ(dψ).

Assuming that f_{p} 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\δ_{y}, dϕ)f_{p}(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∈R^{d}. We find it convenient (see [24]) to express stationarity in
terms of an abstract measurable flowθ :R^{d}×Ω→Ω for which the bijectionsθ_{x} : Ω→Ω,
x∈R^{d}, defined by θ_{x}(·) :=θ(x,·) satisfy the flow propertyθ_{x}◦θ_{y} =θ_{x+y} for allx, y ∈R^{d}.
We then assume that the probability measure P is invariantunder all θ_{x} and that

Ξ(ω)−x= Ξ(θ_{x}ω), x∈R^{d}.

The invariance property (2.3), which is also enjoyed by the additive extensions of the
support measures, implies that the random measures C_{j}^{+}(Ξ,·), j = 0, . . . , d−1, inherit
stationarity from Ξ, i.e.

C_{j}^{+}(Ξ(ω),(A+x)×C) =C_{j}^{+}(Ξ(θ_{x}ω), A×C), (5.1)

for all measurable A, C ⊂R^{d}. In particular it follows that the intensity measures Λ^{+}_{j} are
of product form if Λ^{+}_{j}(· ×R^{d}) is locally finite. Assuming that the intensity

λ^{+}_{j} :=E

C_{j}^{+}(Ξ,[0,1]^{d}×R^{d})
is finite, we have

Λ^{+}_{j}(d(x, b)) = λ^{+}_{j} H^{d}(dx)R_{j}(db),
where R_{j} is a probability measure on R^{d}. In fact, we find that

R_{j}(·) = (λ^{+}_{j})^{−1}Λ^{+}_{j} ([0,1]^{d}× ·)

ifλ^{+}_{j} >0. Note thatR :=R_{d−1}is the rose of directions introduced in the previous section.

Due to stationarity, the volume fraction ¯p := P(x ∈ Ξ) and the contact distribution function

H_{B}(t, A) :=P(d(x)≤t, u(x)∈A|x /∈Ξ)

are independent of x for all measurable A ⊂ R^{d}. Therefore, in the present stationary
situation Corollary 4.6 implies that

∂

∂t t=+0

(1−p)H¯ _{B}(t, A) = 2λ^{s,+}_{d−1}
Z

1{∇hBˇ(u)∈A}hBˇ(u)R^{s}_{d−1}(du).

Our next aim is to introduce stochastic kernels κ_{j}, j = 0, . . . , d−1, from Ω×R^{d} to
R^{d} satisfying

C_{j}^{+}(Ξ, d(z, b)) =κ_{j}(z, db)C_{j}^{+}(Ξ, dz×R^{d}) P −a.s. (5.2)
and being stationary in the sense that

κ_{j}(ω, z,·) = κ_{j}(θ_{z}ω,0,·), (ω, z)∈Ω×R^{d}. (5.3)
To reach that goal we take a measurable setD⊂R^{d} with positive and finite volume. By
(5.1) and the assumption of invariance,

Q_{j}(·) := 1
H^{d}(D)

Z Z

1{z ∈D}1{(θ_{z}ω, b)∈ ·}C_{j}^{+}(Ξ(ω), d(z, b))P(dω) (5.4)
defines a measure on Ω×R^{d}which is independent ofD. In the terminology of Mecke [23],
Qj(· ×A) is the Palm measure of the stationary random measure C_{j}^{+}(Ξ,· ×A), where
A⊂R^{d} is measurable. If λ^{+}_{j} >0, then

Pj(·) := (λ^{+}_{j} )^{−1}Qj(· ×R^{d})

is thePalm probabilityofC_{j}^{+}(Ξ,·×R^{d}). Just for completeness we defineP_{j} :=P ifλ^{+}_{j} = 0.

As in [34] we might interpret P_{j} as a conditional probability measure given that 0 is a
typical point of C_{j}^{+}(Ξ,· ×R^{d}). Equation (5.4) implies the refined Campbell theorem

Z Z

1{(ω, b, x)∈ ·}Q_{j}(d(ω, b))H^{d}(dx) =
Z Z

1{(θ_{z}ω, b, z)∈ ·}C_{j}^{+}(Ξ(ω), d(z, b))P(dω).

(5.5)

Next we introduce a stochastic kernel ˜κ_{j} from Ω toR^{d} by disintegrating Q_{j} according to
Q_{j}(d(ω, b)) = ˜κ_{j}(ω, db)Q_{j}(dω×R^{d}).

The definition κ_{j}(ω, z,·) := ˜κ_{j}(θ_{z}ω,·), (ω, z)∈Ω×R^{d}, 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 P_{j}.

Theorem 5.1 For any measurable f : Ω→[0,∞),

where E_{j} denotes expectation with respect to the Palm probability P_{j}.

Proof. Denote byW an arbitrary subset of R^{d} with H^{d}(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} t^{d−j−1}G_{j}((t,∞]×A). (5.7)
The fact that the contact distributionH_{B}(·,R^{d}) 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 G_{j}({0} ×R^{d}) = 0. Hence the value of the density (5.7) for
t= 0 equals 2λ^{+}_{d−1}R(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_{(ξ}_{n}_{,Z}_{n}_{)} and the point process Ψ = Φ(· × K^{d}) 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_{(ξ}_{n}◦θx,Zn◦θx) =

∞

X

n=1

δ_{(ξ}_{n}−x,Zn)

holds for all x∈R^{d}. In other words, (5.1) is satisfied withC_{j}^{+}(Ξ(·),·) replaced by Φ. We
also assume that the intensity

λ_{Ψ}:=E

Ψ([0,1]^{d})

is strictly positive and finite. The intensity measure α of Φ then satisfies
α(d(x, K)) =λ_{Ψ}H^{d}(dx)Q_{0}(dK),

where Q_{0} is a probability measure on K^{d} 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 ∈ R^{d}. We
can then define a K^{d}-valued stochastic process {Z(x) : x ∈R^{d}} by letting Z(x) :=Z_{n} 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)◦θ_{x}, x ∈ R^{d}. Denoting by P_{Ψ}
the Palm probability of the point process Ψ, we have the following version of the refined
Campbell theorem:

λ_{Ψ}
Z Z

1{(ω, x, Z(0))∈ ·}P_{Ψ}(dω)H^{d}(dx) =
Z Z

1{(θ_{x}ω, x, Z(x))∈ ·}Ψ(ω, dx)P(dω).

(5.8)
LetK 7→P_{Ψ}^{K} be a version of the conditional probability P_{Ψ}(·|Z(0) =K). By (5.8),

P_{(x,K}_{)}=P_{Ψ}^{K}◦θ_{x}, (x, K)∈R^{d}× K^{d}, (5.9)
is one possible choice of the Palm probabilities introduced in the previous section, where
we note that Q_{0} = P_{Ψ}(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 ⊂ R^{d} 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 λ_{Ψ}. We write

V¯_{j} :=
for the mean of the total jth Minkowski support measure and

S¯j(·) :=

Z

Cj(K,R^{d}× ·)Q0(dK)
for the mean jth Minkowski surface area measure. Then we obtain

H_{B}(t) :=H_{B}(t,R^{d}) = 1−exp

Of course, the latter equation can also be derived from Proposition 5.4 using Slivnyak’s
theorem P_{Ψ}(Ψ\δ_{0} ∈ ·) = P(Ψ ∈ ·) and taking into account that Φ also under P_{Ψ} is an
independent marking of Ψ.

In the course of the proof of Proposition 4.13 we have especially shown that Z

H^{d}(K+rB^{d})Q_{0}(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 V_{j}(·), 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, Q_{0} is invariant with respect to rotations, then one can use (5.3.24)
from [28] as well as
(independent of x∈R^{d}) 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