Topics in Stochastic Geometry. Lecture 4 The Boolean model

Transcription

1 Institut für Stochastik Karlsruher Institut für Technologie Topics in Stochastic Geometry Lecture 4 The Boolean model Lectures presented at the Department of Mathematical Sciences University of Bath May 2011

2 1. Definition of the Boolean model Definition Let η = {X n : n N} be a Poisson process with intensity λ and Z 0, Z 1,... a sequence of independent random particles with common distribution Q satisfying the integrability assumption EV d (Z 0 + K ) <, K C d. Assume that η and (Z n ) are independent. Then Z = n N(Z n + ξ n ), is called Boolean model with grain distribution Q and typical grain Z 0.

3 Theorem Let η = {X n : n N} be a Poisson process with positive intensity and R 0, R 1,... a sequence of independent and identically distributed non-negative random variables. Define Z = n N B(X n, R n ), where B(x, r) is a ball with centre x R d and radius r 0. Then P(Z = R d ) = 1 iff ER d 0 =. Remark Let Z 0 := (0, R 0 ), where R 0 is as above. Then Z 0 satisfies the required integrability condition iff ER d 0 <.

4 Lemma Let Z be a Boolean model as above. Then, almost surely, any bounded set is intersected by only finitely many of the (secondary) grains Z n + X n. In particular, Z is closed. Convention A Boolean model Z is considered as a random closed set, that is as a random element in the space of all closed subsets of R d equipped with a suitable σ-field.

5 2. The capacity functional Definition The capacity functional of a random closed set Z is the mapping T Z : C d R defined by T Z (C) := P(Z C ), C C d. Theorem The capacity model of a Boolean model Z is given by T Z (C) = 1 exp[ λev d (Z 0 + C )], C C d, where C := { x : x B}.

6 Corollary A Boolean model Z is stationary, that is Z d = Z + x, x R d. Corollary Assume that the typical grain Z 0 of a Boolean model Z is isotropic, i.e. Z 0 d = ϑz0 for any rotation ϑ. Then Z is isotropic.

7 3. Volume fraction and covariance function Definition The volume fraction p of a stationary random closed set Z is defined by p := EV d ([0, 1] d ). An equivalent definition is p := P(x Z ) for any x R d. Corollary The volume fraction of a Boolean model is given by p = 1 exp( λev d (Z 0 ))

8 Definition The covariance function of a stationary random closed set Z is the function C : R d R d R defined by C(x, y) := P({x, y} Z ) = E[1 Z (x)1 Z (y)], x, y R d. Theorem The covariance function of a Boolean model is given by C(x) = 2p 1 + (1 p) 2 exp(λc 0 (x)), x R d, where C 0 (x) := E[V d (Z 0 (Z 0 x)].

9 4. Ergodicity properties Definition A stationary random closed set Z in R d is called mixing if lim P(Z H θ xh ) = P(Z H)P(Z H ), x for any measurable sets H, H of closed sets. Here θ x H := {A + x : A H }, x R d. Remark Any mixing Z is ergodic, that is P(Z H) {0, 1} for all invariant H.

10 Theorem A Boolean model Z is mixing. Remark The assertion of the theorem can be reduced to the limit relation lim (1 T Z (C (C + x)) = (1 T Z (C))(1 T Z (C )), x for all compact C, C R d. For the Boolean model this can be directly verified.

11 Theorem (Spatial ergodic theorem) Assume that Z is an ergodic random closed set and let W R d be a convex and compact set containing the origin in its interior. Let f (Z ) be a measurable function of Z such that E f (Z ) <. Then the limit lim r 1 f (θ x Z )dx V d (rw ) rw exists almost surely and in L 1 (P) and is given by Ef (Z ). Example For a Boolean model Z the limit 1 lim r V d (rw ) rw 1{x Z }dx exists and is given by the volume fraction of Z.

12 Definition Let K R d be compact and convex. The intrinsic volumes of K are the numbers V 0 (K ),..., V d (K ) uniquely determined by the Steiner formula V d (K + rb d ) = d r j κ j V d j (K ), r 0, j=0 where κ j is the (j-dimensional) volume of the Euclidean unit ball B j in R j. Remark V d (K ) is the Lebesgue measure of K. If K has non-empty interior, then V d 1 (K ) is half the surface area of K. Moreover, V 0 (K ) = 1{K }.

13 Remark Using the inclusion-exclusion formula the intrinsic volumes can be extended (uniquely!) to finite unions K of convex and compact sets. Then V d 1 (K ) is still half the surface area of K while V 0 (K ) is the Euler characteristic of K. Theorem Let Z be a stationary ergodic random closed set that is locally a finite union of convex sets. Under suitable integrability assumptions the limits exist P-almost surely. V j (Z rw ) λ j := lim, j = 0,..., d, r V d (rw )

14 Theorem (Miles) If Z is a Boolean model with convex typical grain Z 0, then λ d 1 = λev d 1 (Z 0 )e λev d (Z 0 ). If Z 0 is isotropic, then λ 0,..., λ d 2 can be computed explicitly in terms of e λev d (Z 0 ) and λev i (Z 0 ), i = 0,..., d 1. For instance, we have for d = 2 that ( λ 0 = e λev d (Z 0 ) λ EV 1(Z 0 ) ) π and in the case d = 3 ) λ 0 = e λev d (Z 0 ) (λ λ2 4 EV 1(Z 0 )EV 2 (Z 0 ) + πλ3 48 (EV 2(Z 0 )) 3.

15 5. References O. Kallenberg (2002). Foundations of Modern Probability. Second Edition, Springer, New York. M. Penrose (2003). Random geometric graphs. Oxford University Press, Oxford. R. Schneider and W. Weil (2008). Stochastic and Integral Geometry. Springer, Berlin. D. Stoyan, W.S. Kendall and J. Mecke (1995). Stochastic Geometry and Its Applications. Second Edition, Wiley, Chichester.