Splitting tessellations in spherical spaces

The concept of splitting tessellations and splitting tessellation processes in spherical spaces of dimension $d\geq 2$ is introduced. Expectations, variances and covariances of spherical curvature measures induced by a splitting tessellation are studied using tools from spherical integral geometry. Also the spherical pair-correlation function of the $(d-1)$-dimensional Hausdorff measure is computed explicitly and compared to its analogue for Poisson great hypersphere tessellations. Finally, the typical cell distribution and the distribution of the typical spherical maximal face of any dimension $k\in\{1,\ldots,d-1\}$ are expressed as mixtures of the related distributions of Poisson great hypersphere tessellations. This in turn is used to determine the expected length and the precise birth time distribution of the typical maximal spherical segment of a splitting tessellation.


Introduction
Random tessellations are among the central topics considered in stochastic geometry and constitute a key model for numerous applications, see [27,33] as well as the references cited therein. While random tessellations in Euclidean spaces have been explored in great detail, their non-Euclidean counterparts are much less intensively studied in the literature. On the other hand, in recent years non-Euclidean models in stochastic geometry have attracted increasing attention, in particular in random geometric structures on manifolds, see e.g. [2,3,7,11,12,13,23]. The papers [3,13] deal with spherical convex hulls of random points on half-spheres and the work [23] considers central limit theorems for point process statistics of point processes on manifolds. Further results for tessellations of the d-dimensional unit sphere by great hyperspheres have been obtained in [2] and, more recently, in [11,12], which generalize at the same time some mean value computations for tessellations generated by great circles on the 2-dimensional sphere S 2 in [21].
In the recent paper [7] a new tessellation model of the two-dimensional unit sphere has been introduced. It arises as the result of a recursive cell splitting scheme and can be regarded as the spherical analogue of the STIT-tessellation model that has been studied intensively in Euclidean stochastic geometry in the last decade. While the work [7] focuses on the two-dimensional case and on metric and combinatorial parameters of individual cells, the present paper has a much broader scope and deals with so-called splitting tessellations in higher dimensional spherical spaces. Our focus lies on first-and second-order properties of cumulative functionals that are induced by the family of spherical curvature measures. In addition, we shall describe precise distributions such as the one of the typical cell and the typical spherical maximal segment.
Let us briefly indicate the random recursive construction of the splitting tessellation Y t . The random construction starts with S d as the unique cell. After an exponential waiting time with parameter 1 a uniform random great hypersphere divides S d into two cells. Now, this branching mechanism continues recursively and independently in both newly created cells. Let us describe this cell splitting scheme more formally: 1. Initiation At time zero we put Y 0 := {S d }, τ 0 := 0 and set a counter n to be equal to 1.

Recursion
Suppose that the counter is n ≥ 1 and that a random time τ n−1 and a random tessellation Y τ n−1 have been realized. Generate a random time τ n such that the holding time τ n − τ n−1 has the same distribution as min c∈Yτ n−1 E c with independent exponentially distributed random variables E c with parameter ν d−1 (S d−1 [c]), which in turn has the same distribution as an exponential random variable with parameter c∈Yτ n−1 ). The parameter ν d−1 (S d−1 [c]) is equal to the probability that a uniform random great hypersphere hits the cell c.
If τ n ≤ t, we -randomly pick a cell c n ∈ Y τ n−1 , where each cell c ∈ Y τ n−1 available at time τ n−1 is selected with probability proportional to -choose a great hypersphere S n ∈ S d−1 [c n ], -put Y τn := (c n , S n , Y τ n−1 ), that is, cell c n is split by S n into two subcells, all other cells remain unchanged, -increase the counter n by one and repeat the recursion step.
If τ n > t, output the random tessellation Y τ n−1 .
A corresponding dynamic description, in a bounded observation window, has been the starting point for various investigations of stationary iteration stable (STIT) or more general branching random tessellations in Euclidean space. We refer to [10] for a detailed description of a more general model. In the current work, we take advantage of another description of the continuous time evolution of the random tessellation Y t for t ≥ 0, in terms of martingale properties of the piecewise constant Markov jump process defined by (Y t ) t≥0 and which takes values in the space of tessellations of the d-dimensional unit sphere.
Let us briefly present a rough overview about the content of this paper. We start in Sections 2.1 and 2.2 by recalling some background material from spherical geometry and spherical integral geometry. Tessellations on the sphere are introduced in Section 2.3, where we also formally define the splitting tessellation process by its generator and by using the general theory of pure jump Markov processes. Our key technical devices are the content of Section 2.4. Here, we construct several classes of martingales connected to the splitting tessellation process. A first application of the theory developed there is the computation of the capacity functional of the random set arising as the union of the cell boundaries of a splitting tessellation Y t in Section 3. The capacity functional is one of the most important characteristics associated with a random set. We also use martingale methods and tools from spherical integral geometry, most notably the spherical Crofton formula, to compute the expected sum over all localized spherical curvatures measures. Spherical integral geometric transformation formulas are developed in Section 5.1 and combined with further martingale tools to determine second-order properties of splitting tessellations. In particular, the variance of the total (d − 1)-dimensional Hausdorff measure of the union of all cell boundaries of cells of Y t is computed explicitly. Moreover, in Section 5.2 we determine the covariance structure of all localized spherical curvature measures. One further second-order parameter that might be associated with an isotropic random measure on S d is its spherical pair-correlation function. We formally introduce this concept in Section 5.3 using the concept of Palm measures on homogeneous spaces. A comparison of the second-order parameters of a splitting tessellation on S d with the corresponding parameters of a Poisson great hypersphere tessellation is the content of Section 6. In particular, we compute there the pair-correlation function of the (d − 1)-dimensional Hausdorff measure of the union of the cell boundaries, since we were not able to locate such a result in the existing literature. The final Section 7 is devoted to distributional properties of the cells and the so-called spherical maximal faces of a splitting tessellation. As a technical tool we introduce in Section 7.1 a continuous-time dynamic version of Poisson great hypersphere tessellations on S d , again by using the general theory of Markov jump processes. This point of view is used in Section 7.2 to establish first a relationship between the cell intensity measure of a splitting tessellation and that of a Poisson great hypersphere tessellation. Furthermore, a similar relationship is proved for the intensity measure of the spherical maximal faces of a splitting tessellation, which is represented as a mixture of intensity measures of spherical faces of a Poisson great hypersphere tessellation. This crucial representation is the key tool in Section 7.3, which leads to a representation of the distribution of the typical k-dimensional spherical maximal face of a splitting tessellation Y t as a mixture of the corresponding distributions of faces in a Poisson great hypersphere tessellation with intensity 0 < s < t. Finally, this allows us to determine precisely the expected length as well as the birth time distribution of a spherical maximal segment of a splitting tessellation on S d .
Whenever possible we compare the results we obtain on the sphere with those for STIT-tessellations in R d that are available in the literature. This makes transparent in which situations the results in the curved space S d are similar to corresponding results in the flat case R d and allows us to highlight where significant differences can be observed.

Preliminaries
The study of random tessellations requires a number of results and tools from stochastic geometry, including point processes of particles, random closed sets, and general methods from the theory of stochastic processes. In this section, we recall or introduce the relevant concepts and explain what is needed from spherical convexity and integral geometry.

Basic notions from spherical geometry
We fix d ≥ 2 and consider the d-dimensional unit sphere where · stands for the usual Euclidean norm in R d+1 . On S d we use the spherical (geodesic) distance (x, y) := arccos( x, y ), for x, y ∈ S d , where we write x, y for the Euclidean scalar product of vectors x and y. As usual, we denote the induced Borel σ-field by B(S d ). For s ≥ 0 we write H s for the s-dimensional Hausdorff measure (normalized as in [9, p. 171]) and put By a spherically convex set we understand the intersection of S d with a non-empty Euclidean convex cone in R d+1 . Let K d denote the collection of all non-empty spherically convex sets of S d , whose elements are called spherical convex bodies.  [27,Chapter 6.5] and [12] for background information and further references).
Next we introduce a particular class of spherically convex sets. A spherical polytope is defined as the intersection of S d with a Euclidean polyhedral cone in R d+1 . The latter is defined as the intersection of finitely many closed halfspaces in R d+1 that contain the origin in their boundaries. It is convenient for our purposes to consider also the space R d+1 as a (degenerate) Euclidean polyhedral cone arising from an empty intersection of halfspaces. This way S d , and also subspheres and closed hemispheres become (degenerate) spherical polytopes. By P d ⊂ K d we denote the space of spherical polytopes and equip P d with the trace σ-field B(P d ) of B(K d ) on P d . In what follows, we shall call the d-dimensional elements of P d cells. For a spherical polytope c ∈ P d , which arises as the intersection of S d with a Euclidean polyhedral coneĉ, we say that a subset F ⊂ c is a j-face of c if F is obtained as the intersection of c with a (j + 1)-dimensional Euclidean face of the polyhedral coneĉ, j ∈ {0, . . . , d}. By F j (c) we denote the collection of all j-faces of c.
By a subsphere of S d of dimension k ∈ {0, . . . , d − 1} we understand the intersection of S d with a (k + 1)-dimensional linear subspace of R d+1 . We denote by S k the space of all k-dimensional subspheres of S d , which is equipped with the subspace topology and the natural Borel σ-field B(S k ). We also put S d := {S d }. Together with the action of the rotation group SO(d + 1) on S k , the space S k becomes a compact homogeneous space and as such it carries a unique Haar probability measure, which is denoted by ν k . For example, the measure ν d−1 has the representation

Elements of spherical integral geometry
Let K ∈ K d be a nonempty spherically convex set and fix 0 < r < π/2. By K r we denote the spherical r-parallel set of K, that is, The spherical version of Steiner's formula [27,Theorem 6.5.1] says that H d (K r \ K) can be expressed as where γ(F, c) is the external angle of c at F . In terms of the polar body c * ∈ P d of c and the set N (c, F ) := {y ∈ c * : x, y = 0} with an arbitrary point x in the relative interior of F , γ(F, c) can be written as For later purposes, we need the following values. If K = xy with (x, y) < π is the unique spherical segment connecting two points x, y ∈ S d , we have V 0 (K) = 1/2 and V 1 (K) = (x, y)/(2π), while V 2 (K) = . . . = V d−1 (K) = 0. In particular, it should be noted that V 0 (K) does not coincide with the Euler characteristic of K, which is in contrast to the Euclidean case. By continuity, these relations extend to the case that (x, y) = π, but then the connecting geodesic is no longer uniquely determined. Moreover, the intrinsic volume of order d − 1 always has the representation where ∂K denotes the boundary of K, provided that K has nonempty interior (for a d-dimensional spherical polytope, the boundary is the union of all of its (d − 1)-dimensional faces). If in addition S ∈ S d−1 , then We can now rephrase one of our crucial devices, namely Crofton's formula for spherical intrinsic volumes. For k ∈ {0, . . . , d} and j ∈ {0, . . . , k}, it states that see [27, p. 261] (note that the case k = d is trivial).
Finally, we recall from [27,Equation (6.63)] that the invariant probability measure of all great hyperspheres hitting a spherically convex set K ∈ K d \ d k=0 S k can be expressed as a sum of spherical intrinsic volumes, that is, Especially for a spherical segment xy with length (x, y) ≤ π we get In the following, we also consider local extensions of the spherical intrinsic volumes. To introduce these, for K ∈ K d and x ∈ S d with (x, K) < π/2, let p(K, x) denote the unique point in K closest to x. For a Borel set A ⊂ S d and 0 ≤ r < π/2, let be the local parallel set of K determined by A and r. Then [27, Theorem 6.5.1] yields that if K ∈ P d is a spherical polytope. In particular, we have Here and in what follows, stands for the restriction of the measure H d−1 to K. (The symbol is generally used to denote the restriction of a measure to a subset.) Again we define φ j (∅, is a finite Borel measure on S d , the jth (spherical) curvature measure of K. For any fixed Borel set A ⊂ S d , the map K → φ j (K, A) is measurable and has the valuation property, that is, is weakly continuous, and φ j is rotation covariant in the sense that φ j (ϑK, ϑA) = φ j (K, A) for all K ∈ K d , A ∈ B(S d ) and ϑ ∈ SO(d + 1), cf. [27,Theorem 6.5.2]. Clearly, we have φ j (K, S d ) = V j (K) and if K ∈ K d has nonempty interior. While (2.2) does not have a local analogue, the spherical Crofton formula extends to curvature measures in the form for all K ∈ K d , Borel sets A ⊂ S d , k ∈ {0, . . . , d} and j ∈ {0, . . . , k}, see [27, p. 261]. Since φ j (K ∩ S, ·) is concentrated on K ∩ S, the integrand in (2.7) can be replaced by φ j (K ∩ S, A) without changing the integral.

Spherical tessellations and spherical splitting tessellations
Spherical tessellations partition the unit sphere into finitely many non-overlapping d-dimensional spherical convex bodies. As in the Euclidean setting, these are necessarily spherical polytopes. For this reason, in the following definition we can equivalently consider finite collections of spherical convex bodies or spherical polytopes (cells).
Definition 2.1. By a tessellation T of S d we understand a finite collection of d-dimensional spherical polytopes in P d such that (ii) any two elements of T have disjoint interiors.
The set of all tessellations of S d is denoted by T d .
In the following, we only need a measurable structure on T d . This will be introduced next. Recall that E = K d is a compact Hausdorff space with countable base. Let N s (E) denote the set of simple counting measures on E, and let F lf (E) denote the set of locally finite (hence finite) subsets of E. We can identify these spaces via the map i s : we have the subspace topology T lf of the Fell topology on F(E), and on N s (E) we consider the vague topology T vg , that is, the coarsest topology such that all evaluation maps η → E g dη are continuous, whenever g : E → R is a non-negative continuous function. The Portmanteau theorem for vague convergence implies that i s is continuous. Hence, i −1 s (T lf ) ⊂ T vg . It is easy to see that this inclusion is strict (think, for example, of two sequences of distinct points (x n ) n∈N and (y n ) n∈N which converge to the same limit point x ∈ E and the sets {x n , y n }, which converges to the set {x} in the Fell topology, but for which the sequence of the associated counting measures does not converge in the vague topology). Let B lf and B vg denote the Borel σ-fields generated by T lf and T vg , respectively. Then we deduce that i −1 s (B lf ) ⊂ B vg , and the same is true for the induced subspace σ-fields on T d and N s ( , which follows from [27, Lemma 3.1.4] (this remains true if g is measurable.) Therefore, we also get the reverse inclusion This equality extends to the intersection with T d so that the measurable structures on T d induced by the vague topology and by the Fell topology coincide. The measurable structure on T d can now be used to define a random tessellation as a measurable map Y : Ω → T d from an underlying probability space (Ω, G, P) into the measurable space Here and in what follows we shall assume that the probability space (Ω, G, P) is rich enough to carry all the random objects we consider in this paper. For T ∈ T d , c ∈ P d and S ∈ S d−1 , we define : , where S ± are the two closed hemispheres determined by S, and otherwise we define (c, S, T ) := T . In other words, if c ∈ T and S ∈ S d−1 [c] then (c, S, T ) is the tessellation arising from T when the cell c is split by the great hypersphere S. It is easy to see that is Borel measurable (with respect to the induced subspace topologies).
The cell-splitting operation can be used to define spherical splitting tessellations in the following way. This point of view had previously been adopted in [10,28,29,30,31,32]. Definition 2.2. By the splitting process (Y t ) t≥0 with initial tessellation Y 0 := {S d } we understand the continuous time pure jump Markov process on T d whose generator A is given by  In these texts, for instance, the existence (by explicit construction) [4, Chapter 15, Section 6] and uniqueness [4,Proposition 15.38] of pure jump processes with a given generator are established. In our applications, we shall exclusively consider time-homogeneous and non-explosive pure jump processes.
Recall that N s (T d ) = i −1 s (T d ) and put µ T := i −1 s (T ) for T ∈ T d . Then we can express the generator A in the form we obtain a probability (rate) kernel π(T, ·) := λ(T ) −1 q(T, ·) and interpret λ as an intensity (rate) function. Then we also have A(T, ·) = q(T, ·) − λ(T )δ T , that is, where If := f . It is important to emphasize that in the present setting the intensity function λ is unbounded.
Without the assumption of a bounded intensity function, the set of all f ∈ F b (T d ) for which (2.8) holds, will be a subset of F b (T d ) which is denoted by D(A) and is called the domain of the generator A. These statements hold for any jump process (X t ) t≥0 with generator L taking values in a Borel space E. While D(A) = F b (T d ) for bounded λ, in our applications the intensity function will be unbounded, and therefore we only know the inclusion D(A) ⊂ F b (T d ). This is the reason why we consider a localization with respect to the values of the intensity function λ in the following.
Remark 2.5. Splitting tessellations in Euclidean spaces have been introduced in [22], where instead of the probability measure ν d−1 a non-normalized version was used. This corresponds to a time change in the splitting process (Y t ) t≥0 . Thus, this choice only affects dimension-dependent constants.

Auxiliary martingales
We use the theory of Markov processes to introduce some classes of martingales associated with the splitting process (Y t ) t≥0 . We start with a preparatory lemma, which is taken from [5, Proposition (14.13), p. 31] to which we also refer for the general definition of a generator of a Markov process and its domain (see [14,Lemma 19.21] for the same result under more restrictive assumptions on the state space and the process). The additional assertion concerning jump processes with bounded intensity functions follows from Remark 2.4. Let us recall from [5, p. 23] that by a Borel space E we understand a topological space which is homeomorphic to a Borel subset of a complete separable metric space.
Lemma 2.6. Let E be a Borel space and let (X t ) t≥0 be a Markov process with values in E and with generator L whose domain is D(L). Further, let f ∈ D(L). Then the random process is a martingale with respect to the filtration induced by (X t ) t≥0 . If (X t ) t≥0 is a jump process with bounded intensity function, then F b (E) = D(L).
In our first application of Lemma 2.6, the space E is the space T d of tessellations of S d . In order to fit into the framework of Lemma 2.6, we remark that T d is a Borel subset of F(K d ) with the Fell topology (see [27,Section 10.1] and the discussion in Section 2.3). The latter is a compact Hausdorff space with countable base (see [27,Theorem 12.2.1]) and hence a Polish space. Further, L will be the generator A from Definition 2.2, and (X t ) t≥0 will be the splitting tessellation process (Y t ) t≥0 . Since we do not know whether D(A) = F b (T d ) in this setting and since we consider functionals f = Σ φ which are unbounded, some localization seems to be unavoidable.
The next result is (for instance) an analogue to [31,Proposition 2]. We present a detailed proof in order to fix some inaccuracies in previous proofs. In what follows, we shall write , t ≥ 0, for the σ-field generated by the splitting process until time t, and Y := (Y t ) t≥0 for the corresponding filtration.
Proposition 2.7. Let φ : P d → R be bounded and measurable, and define Then the stochastic process is a martingale with respect to Y.
Remark 2.8. In this paper all equalities or inequalities involving random variables are implicitly meant to hold almost surely, referring thereby to a common underlying probability space on which all our random objects are defined.
Proof of Proposition 2.7. To see that Σ φ is measurable, we extend φ to K d by setting φ(c) := 0 for c ∈ K d \P d . Then the extension remains bounded and measurable. Let B ∈ B(R) and observe that Since φ is assumed to be bounded, α := sup{|φ(c)| : c ∈ P d } < ∞. For N ∈ N, we consider the truncated functional which is measurable and bounded. In addition to can be constructed on the same probability space (see [8,Chapter 4], [14,Chapter 12]). Let J k , k ∈ N, be the time of the kth jump of (Y t ) t≥0 with the convention that J 0 := 0. By construction we have has a bounded intensity function and Σ N φ is also bounded, the process To relate M N t (φ) to M t (φ), for t ≥ 0, we introduce the almost surely finite random variable Then, clearly we have for t ≥ 0, τ N is a stopping time with respect to both filtrations, Y and Y N . Moreover, by the optional stopping theorem, , for t ≥ 0, also defines a martingale with respect to Y N (see [34,Theorem 7.1.15]).
Thus we obtain This shows that (M τ N t (φ)) t≥0 is an Y N -martingale. Using that τ N is an Y N -stopping time and that Y N s = Y s for s < τ N , it is easy to check that then (M τ N t (φ)) t≥0 is also an F-martingale. Since τ N → ∞, as N → ∞, we conclude that (M t (φ)) t≥0 is a local Y-martingale with respect to the localizing sequence (τ N ) N ∈N . We next argue that the local Y-martingale (M t (φ)) t≥0 is in fact a (proper) martingale by showing that it is of class DL (see [16,Definition 4.8 and Problem 5.19(i)]), that is, for each a > 0 the family (M τ (φ) : τ is a stopping time with τ ≤ a almost surely) is uniformly integrable. For this it is sufficient to prove that for each a > 0, To verify (2.9) we note that, almost surely, As in the Euclidean case one shows that for any a ≥ 0 the number of cells of |Y a | is bounded by a random variable that admits finite moments of all orders, see [22,Lemma 1]. Thus, using the triangle inequality we find that This completes the argument.
To deal below also with second-order properties of splitting tessellations, we first note another consequence of Lemma 2.6 which can be proved in essentially the same way.
2} be bounded and measurable, and define Then the stochastic process Proof. The argument is analogous to the one for Proposition 2.7. Therefore we merely point out the relevant modifications. Since φ i is bounded, which is measurable and bounded. The truncated jump process (Y N t ) t≥0 and related quantities (filtrations, stopping times) are defined as before. Then is an Y N -martingale, which can be related to M t (φ 1 , φ 2 ) as in the preceding proof. When showing that (M t (φ 1 , φ 2 )) t≥0 is indeed a proper martingale (not just a local martingale), we use that E|Y a | 2 < ∞ and the bounds and from which we conclude that the moment condition corresponding to (2.9) is satisfied.
It is clear that the same approach yields corresponding martingale properties of a variety of functionals. In order to deal also with covariances of functionals of splitting tessellations, we next consider the family of time-augmented martingales. We write C 1 ([0, ∞)) for the set of all real-valued continuously differentiable functions on [0, ∞), and C 1 0 ([0, ∞)) ⊂ C 1 ([0, ∞)) for the subset of functions with compact support.
Lemma 2.10. Let F be a Borel space and consider E := F × [0, ∞). Let (X t ) t≥0 be a Markov process with values in F and generator L whose domain is D(L). Then the random process ( X t ) t≥0 with X t = (X t , t) is a Markov process in E. Its generator L is such that
Proof. Since F and [0, ∞) are Borel spaces, E is also a Borel space. That ( X t ) t≥0 is a Markov process is clear. That its generator L is given by (2.10) can easily be confirmed for functions g ∈ D(L) ⊗ C 1 ([0, ∞)), using the definition (see [5, p. 28]) of a generator (see also [5,Section (31.5)]). Finally, we apply Lemma 2.6 to ( X t ) t≥0 to conclude the martingale property of (N t (g)) t≥0 .
A version of Lemma 2.10 for jump processes with bounded intensity function is contained in [8,Theorem 4.4]. The state space there can be extended to a general Borel space. Moreover, minor adjustments of the arguments there show that the result indeed holds for all functions g which are measurable and bounded and such that ∂g ∂s is also bounded. However, this will not be needed in the following.
From the previous lemma we shall derive martingale properties which are adjusted to the subsequent applications involving certain geometric functionals of the cells of the tessellations. The following proposition is the spherical analogue of [30, Equation (7.2)]. Again we provide an argument, since previous arguments require some corrections.
Proof. Note that Using this, the fact that linear combinations of martingales are martingales, the linearity of the generator and Proposition 2.9, it remains to show that is a Y-martingale, where φ : P d → R is bounded and measurable and v ≥ 0 (we use the convention Localizing Σ φ and (Y t ) t≥0 as in the proof of Proposition 2.7, we can apply Lemma 2.10 to the jump process (Y N t ) t≥0 , N ∈ N, which has bounded intensity function and hence its generator has the full domain. Proceeding further as in the proof of Proposition 2.7, we first obtain that (K t (Σ φ )) t≥0 is a local martingale and then a martingale.
The following is a special case (confer [30, Proposition 3.2] for a Euclidean counterpart).
Corollary 2.13. Let φ : P d → R be bounded and measurable, let b ≥ 0 and v ≥ 0, and define Then the random process N t (Ψ φ ) given by (2.11) with (X t ) t≥0 replaced by (Y t ) t≥0 and L by A is a Y-martingale.
Remark 2.14. The results presented in this section do actually not use that the splitting tessellation process takes values in the space of tessellations of the sphere and as such they carry over to the Euclidean set-up as well. In this form the proofs presented here fix a number of technical inaccuracies in the earlier works [30,31] about iteration stable (STIT-) tessellations.

The capacity functional
We fix t ≥ 0 and consider the splitting tessellation Y t of S d with time parameter t. It is convenient for us to consider the random set which consists of the union of all cell boundaries ∂c of cells c in Y t . In particular, Z 0 = ∅. We shall show next that Z t is a random closed subset of S d in the usual sense of stochastic geometry. We recall that a random closed set in S d is a measurable map from an underlying probability , where the Borel σ-field is based on the Fell topology on F(S d ). Moreover, we show the crucial property that Z t is isotropic, that is, Z t has the same distribution as the rotated random set Z t for all ∈ SO(d + 1).
Proof. By construction and by the definition of the required σ-fields in Section 2.3, we know that the map Y t : is also measurable (here δ c denotes the point mass in c ∈ K d , the sum extends over a finite set of spherically convex bodies, and B * vg denotes the Borel σ-field induced by the vague topology on N s (F(S d ))). This shows that ∂ where the sum and the union extend over the same finite set of F ∈ F(S d ) (in fact, the proof of [27, Theorem 3.6.2] carries over to the sphere). Composing these measurable maps yields the assertion.
Next, we show that Z t is isotropic. To verify this, we prove that Y t is isotropic, which means that Y t has the same distribution as Y t for all ∈ SO(d+1). Here, we write T = { c : c ∈ T } for the rotated tessellation T ∈ T d . Recall the definition of the generator A of the splitting tessellation process (Y t ) t≥0 from the previous section. For a bounded and measurable map f : On the other hand, if A denotes the generator of the jump process ( Y t ) t≥0 , the usual definition of the generator involves a uniform limit in T ∈ T d . However, for the following analysis a pointwise limit as considered in [4,Equation (15.21)] is sufficient. In this sense, we have Hence, combining (3.2) and (3.3) we conclude that (Af )(T ) = (A f )(T ) for all functions f and all T ∈ T d , which shows that the generators (as defined in [4]) of (Y t ) t≥0 and ( Y t ) t≥0 coincide.
Remark 3.2. The isotropy of Z t could also be proved by using [4,Proposition 3.39] and induction (over N ). In any case, the rotation invariance of ν d−1 and the rotation covariance of the construction are the crucial points. However, the preceding proof more generally shows that the distribution of Y t is rotation invariant for each t ≥ 0.
The most basic quantity associated with a random closed set is its capacity functional. We are interested in the capacity functional of Z t defined by where C(S d ) is the system of closed subsets of S d . In other words T t (C) is the probability that the compact test set C is hit by the random set Z t . We shall first compute the value of in the case that the set C is connected. This constitutes a direct generalization of [7, Theorem 3.5], but some adjustments of technical details are necessary.
In particular, if C = xy is a spherical segment connecting x, y ∈ S d with (x, y) ≤ π, then Proof. Let t ≥ 0 be fixed. The assertion is apparently true if C = ∅. Hence we assume C = ∅ in the following.
The map φ : P d → R given by φ(c) := 1(C ⊂ c) for c ∈ P d is measurable and bounded. Hence Proposition 2.7 shows that the random process is a martingale with respect to the natural filtration Y induced by (Y t ) t≥0 . Here, c ∩ S + and c ∩ S − are the two sub-cells of c generated by the intersection of c with the great hypersphere S ∈ S d−1 [c]. Let ξ t := c∈Yt 1(C ⊂ c), which is a random variable with values in N 0 .
Claim. ξ t = 1(Z t ∩ C = ∅) holds P-almost surely. In particular, ξ t ∈ {0, 1} almost surely. If ξ t = 1, then almost surely C is contained in the interior of a unique cell of Y t .
. So we restrict ourselves to the cases where Y t = S d and therefore |Y t | ≥ 2.
Step 1: This implies that C ⊂ Z t . Let z 0 ∈ C = ∅ be arbitrarily fixed. Hence by Lemma 3.1 we have for each ∈ SO(d + 1). Using Fubini's theorem and writing ν for the Haar probability measure on SO(d + 1), we deduce that since the boundary of each cell, and hence also Z t , has H d -measure zero. This shows that ξ t ∈ {0, 1} holds P-almost surely.
Step 2: If ξ t = 0, then Z t ∩ C = ∅, since otherwise {C ∩ int(c) : c ∈ Y t } yields a decomposition of C into two non-empty relatively open subsets of C, which contradicts the assumption that C is connected. Hence, in this case we conclude that 1(Z t ∩ C = ∅) = 0.
Step 3: If ξ t = 1, then there is exactly one cell c ∈ Y t such that C ⊂ c. The spherically convex hull (i.e., the positive hull intersected with the sphere) C of C is contained in c, and hence in a hemisphere since |Y t | ≥ 2. Let H t denote the finite (but random) collection of random great hyperspheres arising in the evolution of the jump process up to time t. If H ∈ S d−1 , we say that H supports C if C is contained in one of the two hemispheres determined by H and C ∩ H = ∅. Note that if H supports C, then lin(H) ∈ G(d + 1, d) supports the Euclidean convex body C := conv({o} ∪ C) ⊂ R d+1 and C ∩ lin(H) contains a segment. If C ⊂ c and C ⊂ int(c) for some c ∈ Y t , then the same is true for C in place of C and therefore there is some H ∈ H t which supports C. Since Y t , and hence also H t , has a rotation invariant distribution (see Remark 3.2), we obtain Using Fubini's theorem, we thus get For each realization, we have This finally shows that if ξ t = 1, then Z t ∩ C = ∅ is satisfied P-almost surely, and then the unique cell which contains C already contains C in its interior.
Step 4: Since C is connected, this implies that C ⊂ int(c) for exactly one of the cells c ∈ Z t , in particular, ξ t = 1.
This finally proves the claim and we can continue with the proof of Theorem 3.3.
Since φ(S d ) = 1(C ⊂ S d ) = 1, we thus deduce that Fix s ∈ [0, t] and observe that if ξ s = 0, that is, if there is no cell c ∈ Y s satisfying C ⊂ c, then the integrand of the inner integral is equal to zero. The preceding claim then implies that, if the expression under the expectation is multiplied with ξ t , then the expectation does not change. If ξ s = 1, then almost surely there is a unique cell c 0 ∈ Y s with C ⊂ int(c 0 ). Hence, by the claim, almost surely the expression under the expectation is equal to To justify the second equality, we distinguish two cases. If S ∩ C = ∅, then either C ⊂ int(S + ) or C ⊂ int(S + ), which yields the required equality of the integrands. On the other hand, if S ∩ C = ∅, excluding a set of S ∈ S d−1 [c 0 ] of ν d−1 -measure zero, we deduce that C ⊂ S + and C ⊂ S + , by the argument for (3) in the proof of the claim. This again yields the equality of the integrands for So, we find that and hence Together with the initial condition U 0 (C) = P(Z 0 ∩ C = ∅) = 1, this equation is easily seen to have the unique solution This concludes the first part of the proof. The second part is a direct consequence of (2.3).
Let us now turn to the case where the set C has more than one connected component. In this situation one can find a recursion formula for U t (C). For d = 2 this has been shown in [7,Theorem 3.5], see also [22,Lemma 4] for the STIT-model in the d-dimensional Euclidean space R d . Since precisely the same proof carries over to higher dimensional spherical spaces, we do not provide the details. To present the result we need to introduce some further notation. For a set B ∈ B(S d ), we denote by B the spherical convex hull of B, that is, the intersection of the positive hull of B with S d . Moreover, for two sets be the set of great hyperspheres that separate B 1 and B 2 .
Theorem 3.4. Let C ∈ C(S d ) be such that for some m ∈ N, C = C 1 ∪ . . . ∪ C m with pairwise disjoint connected subsets C 1 , . . . , C m ∈ C(S d ). Then where the sum extends over all partitions Z 1 = i∈I C i , Z 2 = i∈{1,...,m}\I C i , where ∅ = I ⊂ {1, . . . , m} is a proper subset. Theorem 3.3 and Theorem 3.4 provide a description of the capacity functional T t (resp. U t ) of the random closed set Z t on the class of sets consisting of finite unions of pairwise disjoint connected subsets of S d . We remark that this class of subsets of S d is in fact a separating class, that is to say, it is rich enough to determine the capacity functional T t (C) uniquely for all C ∈ C(S d ).
Remark 3.5. Theorem 3.3 and Theorem 3.4 together imply that the capacity functional T t of Z t satisfies T t ( C) = T t (C), C ∈ C(S d ), for all ∈ SO(d + 1). This is consistent with the isotropy of Z t proved in Lemma 3.1. Note, however, that the isotropy of Z t (and Y t ) was used in order to establish Theorem 3.3.

Expected spherical curvature measures
In this section we consider the expectation of the sum of all localized spherical intrinsic volumes, where the sum runs over all cells of a splitting tessellation with time parameter t ≥ 0. Formally, we define for t ≥ 0, j ∈ {0, . . . , d} and A ∈ B(S d ) the random variables The next theorem provides an exact formula for the expectation of Σ j (t; A). More generally, we will consider the following set-up. Let h : S d → R be bounded and Borel measurable. For a finite Borel measure µ on S d , we write In particular, this notation will be applied in writing φ j (c, h), Σ j (t; h) and H d (h). We notice that Σ j (t; h) reduces to Σ j (t; A) for the special choice h = 1 A with A ∈ B(S d ).
where h : S d → R is bounded and measurable.
Proof. The case j = d is obviously true, hence let j ∈ {0, . . . , d − 1} and A ∈ B(S d ). Using that V j (S d ) = φ j (S d , ·) = 0 for j ∈ {0, . . . , d − 1} and the martingale property stated in Proposition 2.7, with the bounded and measurable functional φ(c) = φ j (c, h), c ∈ P d , we see that the random process is a Y-martingale (that is, a martingale with respect to the filtration induced by the splitting process (Y t ) t≥0 ). The valuation property (2.5) of the localized spherical intrinsic volumes yields that Thus, taking expectations in (4.1) and applying the local spherical Crofton formula (2.7), we deduce that Continuing this recursion until we eventually reach the functional Σ d , and using Fubini's theorem, we arrive at Thus it remains to compute EΣ d (s, h). However, with probability one, since Y s is almost surely a tessellation for each s. This immediately implies that which completes the proof.
A quantity of particular interest is the total (d − 1)-dimensional Hausdorff measure of all great hyperspherical pieces that have been constructed by the splitting process up to time t within a set A ∈ B(S d ). Formally, we define where we recall from (3.1) that Z t is the random closed set induced by the splitting tessellation Y t . Using Theorem 4.1, we can easily compute the expectation of H d−1 (t; A). More generally, we compute the expected h-weighted total Hausdorff measure of Z t .
where h : S d → R is bounded and measurable.
First-order properties, that is expectations, of Euclidean intrinsic volumes associated with STITtessellation in a bounded window in R d have been studied in [28,29]. The more general case of localized intrinsic volumes has not been investigated in the Euclidean setting. A comparison of these results with Theorem 4.1 and Corollary 4.2 shows that -up to dimension dependent constants (see Remark 2.5) -the results for STIT-tessellations in R d and splitting tessellations of S d are the same. This means that first-order properties are not sensitive enough to 'feel' the curvature of the underlying space. This will change with the analysis of second-order parameters in the next section.

Variances and covariances
After having investigated the expectation of the functionals Σ j (t; h), for t ≥ 0 and a bounded measurable function h : S d → R, our next goal is to analyse their variances as well as the covariances of Σ i (t; h) and Σ j (t; h) for i = j. We shall start with Σ d−1 (t; h) and then turn to the general case.

The variance of Σ d−1 (t; h)
As described above, the principal goal of this section is to establish a formula for the variance of Σ d−1 (t; h). The result will be based on the following spherical integral geometric transformation formula of Blaschke-Petkantschin type. In principle, such a result could be derived from the very general kinematic formulas in [1,35] that have been obtained using tools from geometric measure theory. However, we prefer to give an elementary and direct proof, which is based on the linear Blaschke-Petkantschin formula in Euclidean spaces (for which an elementary proof is available).
Observe that the integrals on both sides of (5.1) are well defined, but possibly they are both infinite. However, since the left-hand side is finite if g is bounded, the same is true for the integral on the right-hand side (in spite of the unbounded integrand sin( (x, y)) −1 , which remains undefined on a set of measure zero).
Proof. We use the linear Blaschke-Petkantschin formula in R d+1 from [27, Theorem 7.2.1] with d replaced by d + 1 there. In our notation it says that and if x i = o for some i ∈ {x 1 , . . . , q}, then we define g as zero (say). By our assumption on h 1 , . . . , h q and using spherical coordinates in R d+1 and L ∈ G(d + 1, q), respectively, for the left-hand side of (5.2) we get while for the right-hand side we obtain since ∇ q (s 1 u, . . . , s q u q ) = s 1 · · · s q ∇ q (u 1 , . . . , u q ). This proves the formula.

(5.4)
We now apply (5.3) once again, but this time with k = 0, with d replaced by d − 1 and with S d replaced by some fixed great hypersphere S ∈ S d−1 . This gives where S 1 [S] denotes the set of all elements T ∈ S 1 satisfying T ⊂ S. Note that for d = 2 this holds trivially and for d ≥ 3 we have d − 1 ≥ 2 so that (5.3) can indeed be applied. For given S ∈ S d−1 , we write ν S 1 for the invariant Haar probability measure on the space of all T ∈ S 1 with T ⊂ S ∈ S d−1 . Integrating over all S ∈ S d−1 , we thus get where we used the relation which completes the proof.
We are now in the position to derive a formula for the variance of Σ d−1 (t; h) and thus especially for the total h-weighted Hausdorff measure H d−1 (t; h) of the splitting tessellation Y t (for the notation, recall (4.2)).
Theorem 5.4. If t ≥ 0 and h : S d → R is bounded and measurable, then and, in particular, Proof. We definē where Theorem 4.1 was used. For T ∈ T d and t ≥ 0 we put where the valuation property (2.5) of φ d−1 is used for the second equality. Since Crofton's formulas (2.7) yields we conclude that the random process is a Y-martingale. Taking expectations in (5.6) yields that Now, applying (2.4) twice together with Fubini's theorem, we get where Z s is the random closed set induced by the random tessellation Y s (see (3.1)). By Theorem 3.3 and (2.3), we have P(xy ∩ Z s = ∅) = exp − 1 π (x, y)s , and hence Using Fubini's theorem and subsequently carrying out the integration with respect to the time coordinate, we arrive at Note that the right-hand side is finite for non-negative and bounded functions h. This implies the integrability needed for applying Fubini's theorem in the preceding argument. This expression is now transformed by means of the Blaschke-Petkantschin type identity (5.1), which shows that and completes the proof of the first assertion. The second relation easily follows from the first one, since (2.1) and (4.2) yield . This completes the proof.
As a simple consequence, we get the following corollary, which will be used subsequently.
and, in particular, Proof. We apply Theorem 5.
and also expand the right-hand side.

Further variances and covariances
Our next goal is to compute the variance of Σ j (t; h) and, more generally, the covariances of the functionals Σ i (t; h 1 ) and Σ j (t; h 2 ). In order to present the result, we need to introduce some notation. First, let us define for f : [0, t] → R and n ∈ N the iterated integral whenever this is well defined. Moreover, for i, j ∈ {0, . . . , d − 1} we put where here and in what follows h 1 , h 2 : S d → R are bounded, measurable functions. We are now in the position to present closed formulas for the variances and covariances in terms of iterated integrals of EA i,j (s; and, in particular, well as the valuation property of the spherical curvature measures φ i , φ j , we obtain that is a Y-martingale. Using again the Crofton formula (2.7) for the spherical curvature measures, we conclude that is a Y-martingale. Taking expectations yields the recursion formula which expresses the covariance of Σ i (t; h 1 ) and Σ j (t; h 2 ) by means of EA i,j (s; h 1 , h 2 ) as well as covariances with one index increased by one. Continuing this recursion, one eventually arrives at covariances formally involving Σ d (t; h i ), i ∈ {1, 2}, which is identically zero. This in turn shows that the recursion terminates after finitely many steps. Arguing now exactly as at the beginning of Section 3.1 of [29], one arrives at the desired formula after a change of variables.
We notice that Theorem 5.6 is related to the variance formula from the previous section. Indeed, putting k = 0 in Theorem 5.6 yields that which is just Equation (5.7). A similar remark applies to Corollary 5.5

Spherical pair-correlation function
The purpose of this section is to compute the spherical pair-correlation function of the (d − 1)dimensional random Hausdorff measure induced by a splitting tessellation on S d . Before we present our result, we shall first introduce the function we are interested in. To this end, we use the concept of Palm distributions in homogeneous spaces (see [25]) and introduce the function as the spherical analogue of the pair-correlation function of a stationary random measure in R d (see Chapter 7.2.2 in [33]).
Let The unit sphere S d is a homogeneous space that can be identified with the quotient SO(d + 1)/SO(d). Clearly, SO(d+1) acts transitively on S d and SO(d) can be interpreted as the stabilizer of the north pole e := (0, . . . , 0, 1) ∈ R d+1 (or any other fixed point of S d ). Following [25], for x ∈ S d we put Θ x := { ∈ SO(d + 1) : e = x}, denote by ν e the unique Haar probability measure on Θ e and define ν x := ν e • −1 x for an arbitrary x ∈ Θ x (the definition can be shown to be independent of the particular choice of x ). We are now prepared to define the Palm distribution of the isotropic random measure M with respect to e by Using the concept of Palm distributions, we introduce the spherical K-function as well as the spherical pair-correlation function of M.
Definition 5.7. Let M be an isotropic random measure on S d with intensity µ ∈ (0, ∞) and Palm distribution P e M . The spherical K-function of M is defined as where B(e, r) = {x ∈ S d : (x, e) ≤ r} is the geodesic ball centred at e and with radius r. If K M is differentiable, then we call the spherical pair-correlation function of M.
In the following it is useful to rewrite the K-function of M. We get d(x, y)) . (5.8) Remark 5.8. In the Euclidean case, the K-function is defined in a similar way. However, the pair-correlation function is defined by if K is differentiable. Since dκ d r d−1 is the surface area of a (d−1)-sphere of radius r in Euclidean space R d , in spherical space we divide by β d−1 (sin r) d−1 , the (d − 1)-dimensional Hausdorff measure of the boundary of a geodesic ball at distance r to e. The additional factor β d arises, since we are working with the normalized Hausdorff measure.
As anticipated above, we next compute the spherical K-function K d,t and the pair-correlation function g d,t (r) of the (d − 1)-dimensional random Hausdorff measure induced by the splitting tessellation Y t on S d . In other words, we consider the random measure M = H d−1 Z t with Z t defined in (3.1).
Theorem 5.9. If t > 0 and r ∈ (0, π), then which extends to arbitrary bounded measurable functions (x, y) → h(x, y) in the usual way.
Thus, using (5.8) and µ = β d−1 t, we find that For a fixed point x ∈ S d we parametrize y ∈ S d by y = x cos ϕ + z sin ϕ with z ∈ S d ∩ x ⊥ and ϕ ∈ [0, π]. Since the Jacobian of this transformation equals (sin ϕ) d−1 , we get Consequently, K d,t (r) is differentiable with respect to r and we have and the proof is complete.

Comparison with Poisson great hypersphere tessellations
We now compare the pair-correlation function related to the splitting process (Y t ) t≥0 at time t (as discussed in Section 5.3) to the pair-correlation function of an isotropic Poisson process of great hyperspheres of S d with intensity t > 0. To describe the model, we let η t be a Poisson process on S d with intensity measure β −1 d tH d (as in [20] we consider η t as a random measure, but still write x ∈ η t provided that η t ({x}) > 0). Also, put Φ t := η t • F −1 , where F : S d → S d−1 is the map given by F (u) := u ⊥ ∩ S d . We denote by Y t the tessellation of S d induced by Φ t and let be the associated random closed set, see Figure 6.1 for an illustration. Let us first determine the intensity µ := EH d−1 (Z t ∩ S d ) of the isotropic random measure H d−1 Z t .
Here, we used that, for all non-negative, measurable f : S d → R, This follows from the fact that both sides of this equation define rotation invariant measures on the unit sphere and hence these measures must be proportional. The constant is easily determined by choosing the function f ≡ 1. Hence, the intensity µ equals The proof is thus complete.
We emphasize that the intensity µ coincides with the corresponding intensity for H d−1 Z t , where Z t is the random set corresponding to a splitting tessellation with parameter t. Next, we determine the K-function K d,t (r) of H d−1 Z t as well as the corresponding pair-correlation function g d,t (r).

Typical cells and faces, and their distributions
In this section, we describe additional relations between spherical splitting tessellations and tessellations generated by Poisson processes of great hyperspheres.

A dynamic Poisson great hypersphere tessellation process
Many of our arguments and results we present below are based on a link between splitting tessellations and Poisson great hypersphere tessellations. To establish this link, we use a continuous-time dynamic version of the latter model and introduce in this section a dynamic Poisson great hypersphere tessellation process. To this end, for a tessellation T ∈ T d and a great hypersphere S ∈ S d−1 we define In other words, the tessellation ⊗(S, T ) is obtained from T by dividing by S all cells of T whose interior has non-empty intersection with the great hypersphere S. This operation is similar to the splitting operation (c, S, T ) in the context of splitting tessellations, where only the single cell c gets divided by S provided S intersects the interior of c.
We define now the continuous-time Markov process (Y t ) t≥0 with initial tessellation Y 0 = {S d } in T d whose generator A is given by where f : T d → R is any bounded and measurable function. Notice that for any t > 0 the random tessellation Y t has the same distribution as a Poisson great hypersphere tessellation with intensity t as defined in the previous section and which was denoted there by the same symbol. For this reason, (Y t ) t≥0 is called a dynamic Poisson great hypersphere tessellation process. Then it is also clear that Y t is isotropic.
In order to provide a proof of these statements, we verify that the generator A of a process of Poisson great hypersphere tessellations with intensity t at time t equals A. For this, let η be a Poisson process on [0, ∞) × S d with intensity measure H 1 ⊗ β −1 d H d and recall the definition of the isotropic Poisson process η t on S d with intensity t (also recall that we regard η t and also η as random measures). Then η t is equal in distribution to η [0,t] := η ([0, t]×S d ). We write Tess(η [0,t] ) for the tessellation generated by {u ⊥ : (r, u) ∈ supp(η [0,t] )} and Tess(T, η [a,b] ), 0 ≤ a ≤ b, for the tessellation obtained from a great hypersphere tessellation T by further intersection with great hyperspheres derived from η [a,b] . Let t ≥ 0, h > 0, and let f : T d → R be bounded and measurable. Since η [0,t+h] has the same distribution as η [0,t] + η (t,t+h] and since η [0,t] and η (t,t+h] are stochastically independent, we get Thus we deduce which yields the equality A = A , since ν d−1 is a probability measure. From the equality of the generators we finally deduce from [4,Proposition 15.38] that the two corresponding tessellationvalued processes are identically distributed. In particular, for any t > 0 the random tessellation Y t has the same distribution as a Poisson great hypersphere tessellation with intensity t as defined in Section 6.

Relationships for intensity measures
We denote by Y t a splitting tessellation with time parameter t ≥ 0. Then we define the random measure M t and its intensity measure M t on P d by Similarly, for a Poisson great hypersphere tessellation Y t with intensity t ≥ 0 we put Repeating the proof of [31, Theorem 1] we obtain the following result. We shall nevertheless provide the argument for completeness and to complement some details in [31] that have been left out.
Proof. Let φ : P d → R be bounded and measurable. Then Proposition 2.7 ensures that is a Y-martingale. Taking expectations and using Campbell's theorem, we get Let us denote by M bv (P d ) the Banach space of real-valued Borel measures on P d with the total variation norm · TV . Further, we write δ p for the Dirac measure at p ∈ P d . Then the linear operator is bounded with operator norm Γ ≤ 3. As observed in the proof of Proposition 2.7, we have M t TV = E|Y t | ≤ E|Y a | =: c a < ∞ if 0 ≤ t ≤ a. Then (7.2) can equivalently be written in the form and hence M t − M r TV ≤ 3c a |t − r| for 0 ≤ r ≤ t ≤ a.
Next, we consider the dynamic Poisson great hypersphere tessellation process (Y t ) t≥0 we introduced in Section 7.1. Proposition 2.7, applied to the Markov process (Y t ) t≥0 with generator A, yields that is a martingale with respect to the filtration induced by (Y t ) t≥0 . In fact, a localization procedure similar to the one used in the proof of Proposition 2.7 first shows that this process is a local martingale. In order to verify that this process is of class DL and hence a proper martingale, one also needs that the moments of |Y t | are finite. However, this is the case since the number of cells can be expressed as a deterministic polynomial of the number of great hyperspheres of η t (see [27,Lemma 8.2.1]), which in turn is a Poisson random variable having finite moments of all orders. In particular, we have M t TV = E|Y t | ≤ c a < ∞ if 0 ≤ t ≤ a. Since the right-hand side in (7.3) is the same as in (7.1), but with Y s replaced by Y s , we also obtain that Hence (M t ) t≥0 and (M t ) t≥0 solve the same initial value problem. In the current situation the solution of this problem is unique (see [6, Section 1]), which implies the assertion. In fact, let 0 ≤ t ≤ a be arbitrary and putc a := c a + c a < ∞.

Iteration of this argument yields
M t TV ≤c a (3t) n n! , 0 ≤ t ≤ a , n ∈ N .
Thus, taking the limit as n → ∞, we conclude that M t TV = 0 for all 0 ≤ t ≤ a, which proves the assertion.
Let T be a (deterministic) splitting tessellation (that is, a tessellation obtained by a successive splitting process). By a maximal spherical face of dimension d − 1 of T we mean the separating pieces of great hyperspheres that arise in the construction of the splitting tessellation T . Further, by a maximal spherical face of dimension k ∈ {0, . . . , d − 2} of T we understand any k-face of a maximal spherical face of dimension d − 1. We denote by F * k (T ) the collection of these maximal spherical k-faces of T , which is a subset of the space P d k of k-dimensional spherical polytopes in S d . We introduce on P d k the random measure F (k) t and its intensity measure F Similarly, for a great hypersphere tessellation T , we understand by a spherical k-face of T any k-face of a cell of T and denote by F k (T ) the collection of all such faces (each k-face is included only once in F k (T ), although it arises as a k-face of precisely 2 d−k cells). On P d k we then define the measures F (k) The next proposition is the analogue of [31, Theorem 2]. Again, the proof is basically the same as in the Euclidean case, but we give the argument for the sake of completeness. tessellation (and as zero otherwise), and arguing as in the proof of Proposition 2.7, we conclude that the random process is a Y-martingale. Taking expectations and using Proposition 7.1, we get Let η t be a Poisson process on S d , as defined in Section 6. Then, for any s ∈ (0, t), by Campbell's theorem (see [27, We thus conclude that Moreover,

Typical spherical maximal faces
This section requires the concept of a typical object of an isotropic random tessellation T on S d (see also [11,17,18,25]). Let X = X (T ) be a class of (possibly lower-dimensional) spherical polytopes determined by T and associate with each x ∈ X a centre z(x) ∈ S d in a measurable way and such that z( x) = z(x) for all ∈ SO(d + 1) (rotation covariance). We assume that ξ X := x∈X δ z(x) defines a simple, isotropic, point process on S d with positive and finite intensity. As in the previous sections, we let e = (0, . . . , 0, 1) be the north pole of S d and denote by P e ξ X the Palm measure of the point process ξ X with respect to e. Under P e ξ X there is an almost surely uniquely determined x ∈ X with z(x) = e. We call such an x the typical object of the class X with respect to the given centre function. For example, X could be the class of cells of an isotropic random tessellation T and the associated typical object is then called the typical cell of T . In particular, we shall use the notation Q for the distribution of the typical cell of a splitting tessellation Y t and that of a Poisson great hypersphere tessellation Y t of intensity t > 0 as considered in the previous section, respectively. Moreover, X could also be the class of spherical maximal k-faces of a splitting tessellation Y t for some t > 0, the corresponding typical object is the so-called typical spherical maximal k-face of Y t . In what follows, we shall use the notation Q To formulate our result we need to introduce some further notation. By N k (t) we denote the expected number of spherical maximal k-faces of the splitting tessellation Y t and, similarly, we write N k (t) for the expected number of spherical k-faces of a Poisson great hypersphere tessellation Y t of intensity t > 0. The computation of N k (t) or N k (t) is in general rather involved as considerations in [2,12] indicate. For this reason we shall carry out explicit computations only for the special case k = 1 below.
This proves the first assertion.
In the following corollaries, we write γ(a, x) := x 0 s a−1 e −s ds, where a, x > 0, for the lower incomplete gamma function. The first corollary provides an explicit description of the distribution of the typical spherical maximal segment (that is, the typical spherical maximal 1-face) of a splitting tessellation on S d as a mixture of distributions of typical spherical edges (that is, typical spherical 1-faces) in a Poisson great hypersphere tessellation. Besides the general formulas we treat especially the case d = 2, which is compared to the results in the Euclidean case at the end of this section. Proof. In view of (7.5) we have to compute the mean values N 1 (s) and N 1 (t). Let ξ(s) be the random number of spherical edges of a Poisson great hypersphere tessellation of intensity s > 0. We denote by P (s) the number of great hyperspheres in Y s , which is a Poisson distributed random variable with parameter s. If P (s) ∈ {0, 1, . . . , d − 2} then ξ(s) = 0. If P (s) = d − 1, then ξ(s) = 1, while for P (s) ≥ d, we have ξ(s) = 2d P (s) d . Indeed, any collection of d great hyperspheres generating Y s (and which are in general position with probability one) induces a pair of antipodal vertices. From each such vertex there are precisely 2d emanating edges and each edge has two vertices as its endpoints (see also [12,Equation (16)  The previous result can be used, in particular, to determine the expected length of the typical spherical maximal segment of a splitting tessellation Y t on S d .
Corollary 7.5. The expected length of the typical spherical maximal segment of a splitting tessellation Y t on S d with t > 0 equals Especially, if d = 2, this reduces to 2πt Proof. Let L(Y s ) be the expected length of the typical edge in a Poisson great hypersphere tessellation of S d with intensity s > 0 and let L(Y s ) be the total edge length of Y s , that is the sum of the lengths of all edges of Y s . We have Thus, using (7.6), the expected length of the typical edge of Y s equals L(Y s ) = EL(Y s ) N 1 (s) = 2π 2s + e −s . 1, t) .

Combining this with Corollary 7.4 yields
The formula for d = 2 follows once again from the observation that γ(1, t) = 1 − e −t .
Finally, let us consider the distribution of the 'birth time' of the typical spherical maximal segment of a splitting tessellation Y t with t > 0. To define this concept formally, we recall the definition of the centre function z from the beginning of this subsection. In the particular case we consider, if x ∈ P d 1 is a spherical line segment, then z(x) is chosen as the midpoint of x. By the continuous time construction of Y t , each spherical maximal segment x of Y t has a well defined birth time β(x). This is the uniquely determined s ∈ (0, t) for which a cell c ∈ Y s− is split by a great hypersphere S ∈ [c] such that c ∩ S = x (here, Y s− stands for the left limit of Y s , that is, the tessellation right before the split). This gives rise to a marked point process ζ t on S d with mark space (0, ∞), where for each t ≥ 0, ζ t consists of all centres (midpoints) z(x) of spherical maximal segments x constructed until time t together with their corresponding birth times β(x) (see [27,Chapter 3.5] for background material on marked point processes). The mark attached to the centre (midpoint) of the typical spherical maximal segment of Y t will be denoted by β(Y t ) and is called the birth time of the typical spherical maximal segment. In the next result we determine the precise distribution of the random variable β(Y t ).
Corollary 7.6. For each t > 0, the random variable β(Y t ) ∈ (0, t) has density s → ds d−2 (2s + e −s ) 2t d + dγ(d − 1, t) with respect to the Lebesgue measure on (0, t). Especially, if d = 2, this reduces to s → 2s + e −s t 2 + 1 − e −t . Proof. The distribution of β(Y t ) is just the mark distribution of the marked point process ζ t , whose distribution function in turn is given by the ratio where N 1 (t) is the expected number of edges of Y t and N 1 (s, t) is the expected number of edges whose midpoints are marked with a birth time less than or equal to s, see [27, p. 84]. By the continuous time Markovian construction of Y t we have that N 1 (s, t) = N 1 (s), the expected number of edges of Y s . Using the formula (7.7) for N 1 (t) (and also with t replaced by s for N 1 (s)) and differentiating the resulting expression with respect to s completes the proof.
The results of the two previous corollaries might be compared with the corresponding situation for stationary and isotropic STIT-tessellations in R d with time parameter t > 0. It is known from [31,Corollary 4] that the expected length of the typical maximal segment of such a tessellation is given by Especially, if d = 2 this reduces to π t , see the left panel in Figure 7.1 for a comparison with the splitting tessellation on S 2 . Moreover, it is known from the discussion after [31, Theorem 3] that for a STIT-tessellation in R d with time parameter t > 0 the birth time distribution of the typical maximal segment has density on (0, t) with respect to the Lebesgue measure given by s → ds d−1 t d , which is the density of a beta distribution on (0, t) with shape parameters d and 1. The right panel in Figure 7.1 shows a comparison of this density with that of β(Y t ).