Tessellation-valued processes that are generated by cell division

Werner Nagel

Friedrich-Schiller-Universität Jena,

Institut für Stochastik,

Ernst-Abbe-Platz 2, 07743 Jena, Germany.

Email<EMAIL_ADDRESS>
Servet Martínez

Universidad de Chile, Departamento Ingeniería Matemática and Centro
Modelamiento Matemático,

UMI 2807 CNRS, Casilla 170-3, Correo 3,

Santiago, Chile.

Email<EMAIL_ADDRESS>
###### Abstract

Processes of random tessellations of the Euclidean space $\mathbb{R}^{d}$,
$d\geq 1$, are considered which are generated by subsequent division of their
cells. Such processes are characterized by the laws of the life times of the
cells until their division and by the laws for the random hyperplanes that
divide the cells at the end of their life times. The STIT tessellation
processes are a reference model. In the present paper a generalization
concerning the life time distributions is introduced, a sufficient condition
for the existence of such cell division tessellation processes is provided and
a construction is described. In particular, for the case that the random
dividing hyperplanes have a Mondrian distribution – which means that all cells
of the tessellations are cuboids – it is shown that the intrinsic volumes,
except the Euler characteristic, can be used as the parameter for the
exponential life time distribution of the cells.

Keywords: Stochastic geometry; stochastic process of random tessellations;
STIT tessellation; crack pattern; Mondrian model; fragmentation

AMS subject classification: Primary 60D05; Secondary 60J25; 60J76; 60J80

##  1 Introduction

The study of random tessellations (or mosaics) is a substantial part of
stochastic geometry. The Voronoi tessellations and Poisson hyperplane
tessellations are classical and well-established. Motivated by modeling
fracture or crack patterns as they are observed for example in geology,
materials science, nanotechnology or drying soil patterns [30, 2, 22, 5, 29],
also space-time processes of tessellations are considered where the cells are
consecutively divided. Recently, such models are also of growing interest in
the context of machine learning, see [23] and the references therein.

Several of the approaches suggested so far, turned out to be cumbersome, if
not unfeasible, with respect to theoretical investigations, and only
simulation studies can be performed. In [20] the STIT tessellation process was
introduced, which is a cell division process allowing for numerous theoretical
results, see for example [21, 16, 17, 25, 27, 26, 14]. Due to its nice
mathematical properties, it can be considered as a reference model for
fracture patterns. On the other hand, some statistical goodness-of-fit checks
indicate the need for modification and adaption of the STIT model, see [19,
11]. This motivates the present paper.

Our work is essentially inspired by Cowan’s paper [3] where a systematic
approach to a wide class of cell division processes is provided by introducing
the ’selection rules’ – in the present paper adapted as ’life time
distributions’ – and ’division rules’.

In [28, 4] a theoretical base for generalizations of the STIT model was
provided. In those papers the focus was on modifications of the division rules
for the cells.

Our purpose is the construction of tessellation-valued cell division processes
in the $d$-dimensional Euclidean space $\mathbb{R}^{d}$, where the life time
distributions of the cells differ from those ones of the STIT model. We
provide sufficient conditions and a rigorous proof for their existence, which
is a critical issue for such models, and we study some of their features.

The idea of the proof of Theorem 1 is based on the ’global construction’ of
STIT tessellations by Mecke in [17]. For this reason, the models which we
consider, all have the same division rule like the STIT tessellation, driven
by a translation invariant hyperplane measure. The new results of this paper
concern a variety of life time distributions of the cells.

In Section 3 we start with a proof of the tail triviality of the zero-cell
process of a Poisson hyperplane tessellation process. Because this process of
zero-cells plays a central role in several proofs of cell division processes,
this is a result of its own interest. Then in Section 4 a sufficient condition
for the existence of a class of cell division processes is provided together
with their construction. Several more concrete results are shown in Section 5
for the particular case of the so-called Mondrian model where all the
tessellation cells are cuboids. We conclude by describing a link between a
particular cell division process in a bounded window and a fragmentation as
defined in [1].

##  2 Notation

By $\mathbb{R}^{d}$ we denote the $d$-dimensional Euclidean space, $d\geq 1$,
with the scalar product $\langle\cdot,\cdot\rangle$ . The origin is $o$, and
$S^{d-1}$ is the unit sphere with the Borel $\sigma$-algebra
${\mathcal{B}}(S^{d-1})$. Denote by ${\sf B}_{r}$ the ball centered in the
origin $o$ with radius $r>0$. The topological interior of a set
$A\subseteq\mathbb{R}^{d}$ is ${\rm int}(A)$. The Lebesgue measure on
$\mathbb{R}$ is denoted by $\lambda$. The set of the nonnegative integers is
$\mathbb{N}_{0}$, $\mathbb{Z}$ denotes the set of integers and
$\mathbb{Z}_{-}:=\\{i\in\mathbb{Z}:\,i\leq 0\\}$. The symbol
$\stackrel{{\scriptstyle D}}{{=}}$ means the identity of distributions of two
random variables. We use the abbreviation ’i.i.d.’ for ’independent
identically distributed’ random variables.

Let ${\cal H}$ denote the set of all hyperplanes in $\mathbb{R}^{d}$, ${\cal
H}$ endowed with the Borel $\sigma$-algebra associated to the Fell topology.
For a hyperplane $h\in{\cal H}$ we use the parametrization

$h(u,x):=\\{y\in\mathbb{R}^{d}:\,\langle y,u\rangle=x\\},\quad u\in
S^{d-1},\,x\in\mathbb{R}.$

Note, that $h(u,x)=h(-u,-x)$, thus all hyperplanes can be parametrized in two
different ways.

Let $\varphi$ denote an even probability measure on $S^{d-1}$, which is a
probability measure on $S^{d-1}$ with $\varphi(A)=\varphi(-A)$ for all
$A\in{\mathcal{B}}(S^{d-1})$.

Throughout this paper we use a translation invariant measure $\Lambda$ on
${\cal H}$ which satisfies

$\int_{{\cal H}}f{\rm
d}\Lambda=\int_{S^{d-1}}\int_{\mathbb{R}}f(h(u,r))\lambda({\rm
d}r)\varphi({\rm d}u)$ (1)

for all nonnegative measurable functions $f:{\cal H}\to\mathbb{R}$. The
probability measure $\varphi$ is called the spherical directional
distribution. We need a further assumption on $\varphi$ which guarantees that
the constructed structures are indeed tessellations with bounded cells, see p.
486 of [24],

$\mbox{$\varphi$ is not concentrated on a great subsphere of }S^{d-1}.$ (2)

A great subsphere is the intersection of $S^{d-1}$ with a hyperplane through
the origin. Condition (2) is equivalent to the assumption that there is no
line in $\mathbb{R}^{d}$ to which $\varphi$-almost-all hyperplanes are
parallel.

A polytope in $\mathbb{R}^{d}$ is the convex hull of a nonempty finite set of
points. By ${{\cal P}_{d}}$ we denote the set of all $d$-dimensional polytopes
in $\mathbb{R}^{d}$.

###### Definition 1

A tessellation of $\mathbb{R}^{d}$ is a countable set $T\subset{{\cal P}_{d}}$
of $d$-dimensional polytopes satisfying the following conditions:

  1. (a)

$\displaystyle{\bigcup_{z\in T}z=\mathbb{R}^{d}}$ (covering),

  2. (b)

If $z,z^{\prime}\in T$ and $z\not=z^{\prime}$, then ${\rm int}(z)\cap{\rm
int}(z^{\prime})=\emptyset$ (disjoint interiors),

  3. (c)

The set $\\{z\in T:z\cap C\not=\emptyset\\}$ is finite for all compact sets
$C\subset\mathbb{R}^{d}$ (local finiteness).

The polytopes $z\in T$ are referred to as the cells of $T$.

Let $\mathbb{T}$ denote the set of all tessellations of $\mathbb{R}^{d}$,
endowed with the usual $\sigma$-algebra, which is associated with the Borel
$\sigma$-algebra for the Fell topology in the space of closed subsets of
$\mathbb{R}^{d}$, using that the union of cell boundaries $\partial
T:=\bigcup_{z\in T}\partial z$, also referred to as the $(d-1)$-skeleton, of a
tessellation $T$ is a closed subset of $\mathbb{R}^{d}$. A random tessellation
is a measurable mapping from a probability space to $\mathbb{T}$.

For more details on the definition of random tessellation we refer to [24].

In this paper, Poisson point processes are used extensively. For this,we refer
to the book [8], in particular to Chapters 5 and 7 there.

##  3 An auxiliary Poisson process of hyperplanes that are marked with birth
times, and the process of zero-cells

Let $\Lambda$ be a translation invariant measure on the space ${\cal H}$ of
hyperplanes in $\mathbb{R}^{d}$, satisfying (1) and (2). We denote by
$\hat{X}^{*}$ a Poisson point process on ${\cal H}\times(0,\infty)$ with the
intensity measure $\Lambda\otimes\lambda_{+}$, where $\lambda_{+}$ is the
Lebesgue measure on $(0,\infty)$. If $(h,t)\in\hat{X}^{*}$, we interpret this
as a hyperplane with birth time $t$. Furthermore, define
$\hat{X}^{*}_{t}:=\\{(h,t^{\prime})\in\hat{X}^{*}:\,t^{\prime}\leq t\\}$ and
$\hat{X}_{t}:=\\{h\in{\cal H}:\exists(h,t^{\prime})\in X^{*}_{t}\\}$. Now we
can introduce the process $(\tilde{z}^{o}_{t},t>0)$, where $\tilde{z}^{o}_{t}$
is the zero-cell – that is the cell containing the origin $o$ – of the Poisson
hyperplane tessellation of $\mathbb{R}^{d}$ induced by $\hat{X}_{t}$. The
process $(\tilde{z}^{o}_{t},t>0)$ is a pure jump Markov process, with and
$\tilde{z}^{o}_{t^{\prime}}\subseteq\tilde{z}^{o}_{t}$ if $t<t^{\prime}$. And,
most importantly,

$\bigcup_{t>0}\tilde{z}^{o}_{t}=\mathbb{R}^{d}\quad a.s.$ (3)

Denote by $(t_{i},\,i\in\mathbb{Z})$ the ordered sequence of jump times of
$(\tilde{z}^{o}_{t},t>0)$ with $t_{0}<1<t_{1}$. Correspondingly,
$(h_{i},t_{i})\in\hat{X}^{*}$ are the time marked dividing hyperplanes causing
a jump of $(\tilde{z}^{o}_{t},t>0)$.

Denote by

$(\tilde{z}^{o}_{(i)},\,i\in\mathbb{Z})$ (4)

the sequence of zero cells with $\tilde{z}^{o}_{(i)}:=\tilde{z}^{o}_{t_{i}}$ .

Consider the process $(\hat{z}_{n},\,n\in\mathbb{N}_{0})$ with
$\hat{z}_{n}:=\tilde{z}^{o}_{(-n)}$, and denote by ${\cal
F}_{n}:=\sigma(\hat{z}_{n})$, $n\in\mathbb{N}_{0}$, the $\sigma$-algebra
generated by $\hat{z}_{n}$ . The tail-$\sigma$-algebra is then

${\cal F}_{\infty}:=\bigcap_{m\in\mathbb{N}_{0}}\bigvee_{k\geq m}{\cal
F}_{k},$

where $\bigvee$ stands for the $\sigma$-algebra generated by the union of the
respective $\sigma$-algebras.

###### Lemma 1

The tail-$\sigma$-algebra ${\cal F}_{\infty}$ of the process
$(\hat{z}_{n},\,n\in\mathbb{N}_{0})$ is trivial, that is
$\mathbb{P}(B)\in\\{0,1\\}$ for all $B\in{\cal F}_{\infty}$ .

Proof Let be $B\in{\cal F}_{\infty}$ and ${\varepsilon}>0$ fixed. Then there
are an $m\in\mathbb{N}$ and a $B_{m}\in\bigvee_{k\leq m}{\cal F}_{k}$ such
that $\mathbb{P}(B\Delta B_{m})<{\varepsilon}$ , see Theorem D, Chapter III of
[6]. In the following we consider $m$ and $B_{m}$ as fixed.

The main idea of the proof is to construct times ${\tilde{t}}<t^{*}$, two
polytopes $W^{\prime}\subset W$ containing the origin $o$, and events $E$,
$E_{m}$, $A$, $A_{n}$, $n>m$, such that

– the event $E\cap E_{m}$ implies that
$W^{\prime}\subseteq\hat{z}_{m}\subseteq W$, and that the jump time
$t_{-m}\geq t^{*}$ ,

– the event $A\cap A_{n}$ implies that $W\subseteq\hat{z}_{n}$, and that the
jump time $t_{-n}\leq{\tilde{t}}$ .

For $W^{\prime},W\in{{\cal P}_{d}}$, $W^{\prime}\subset W$, with the origin
$o$ in the interior of $W^{\prime}$, define the time
$S(W^{\prime},W):=\inf\\{t>0:\,\tilde{z}^{o}_{t}\subseteq W\\}$ which is the
time of first separation of the boundary of $W^{\prime}$ from the boundary of
$W$ by the boundary of the zero-cell, in [13] introduced as the ’encapsulation
time’.

For $t^{*}\in(0,1)$ and $r>0$ define the intervals
$a_{i}:=\left((i-1)\frac{r}{m},\,i\frac{r}{m}\right)$ of length $r/m$, and
time intervals
$\Delta_{i}:=\left(t^{*}+(i-1)\frac{1-t^{*}}{m},\,t^{*}+i\frac{1-t^{*}}{m}\right)$
of length $(1-t^{*})/m$ between $t^{*}$ and 1, $i\in\\{1,\ldots,m\\}$.

Now, for $m\in\mathbb{N}$ and ${\varepsilon}>0$ choose a pair
$(t^{*},r)\in(0,1)\times(0,\infty)$ such that for the event

$\displaystyle E_{m}:=$
$\displaystyle\\{\hat{X}^{*}\cap\left(\\{h(u,x)\in{\cal
H}:\,x\in(-r,r)\\}\times(0,t^{*})\right)=\emptyset\\}$
$\displaystyle\cap\bigcap_{i=1}^{m}\left\\{\hat{X}^{*}\cap\left(\\{h(u,x)\in{\cal
H}:\,x\in a_{i}\cup(-a_{i})\\}\times\Delta_{i}\right)\not=\emptyset\right\\}$

holds

$\mathbb{P}(E_{m})={\rm exp}(-t^{*}r)\,\left(1-{\rm
exp}\left(-\frac{2(1-t^{*})\,r}{m^{2}}\right)\right)^{m}>1-{\varepsilon}\,.$
(5)

This can be realized by choosing
$r>-\frac{m^{2}}{2}\ln\left(1-(1-{\varepsilon})^{\frac{1}{2m}}\right)$ and
then $t^{*}<-\frac{1}{2r}\ln(1-{\varepsilon})$, which guarantees that each of
the two factors is greater than $\sqrt{1-{\varepsilon}}$.

The event $E_{m}$ means that the ball ${\sf B}_{r}$ is not intersected by a
hyperplane of $\hat{X}^{*}$ until time $t^{*}$ and then in each of the $m$
time intervals
$\Delta_{i}:=\left(t^{*}+(i-1)\frac{1-t^{*}}{m},\,t^{*}+i\frac{1-t^{*}}{m}\right)$
of length $(1-t^{*})/m$ between $t^{*}$ and 1 there is at least one hyperplane
of $\hat{X}^{*}$ with a distance to the origin in the interval
$a_{i}:=\left((i-1)\frac{r}{m},\,i\frac{r}{m}\right)$, $i\in\\{1,\ldots,m\\}$.
This guarantees that there are at least $m$ jumps of the zero-cell process in
the time interval $(t^{*},1)$.

The pair $(t^{*},r)$ is now considered to be fixed. Choose
$W^{\prime}\in{{\cal P}_{d}}$ such that ${\sf B}_{r}\subset W^{\prime}$. By
Lemma 5 of of [13] there exists a $W\in{{\cal P}_{d}}$ with $W^{\prime}\subset
W$ such that for the event $E:=\\{S(W^{\prime},W)<t^{*}\\}$ holds

$\mathbb{P}(E)=\mathbb{P}(S(W^{\prime},W)<t^{*})>1-{\varepsilon}\,.$ (6)

From now on, also $W$ is considered to be fixed. Furthermore, choose
${\tilde{t}}\in(0,t^{*})$ such that for the event
$A:=\\{[W]\cap\hat{X}_{{\tilde{t}}}=\emptyset\\}$, which means that at time
$\tilde{t}$ the window $W$ is contained in the zero-cell,

$\mathbb{P}(A)={\rm
exp}\left(-{\tilde{t}}\,\Lambda([W])\right)>1-{\varepsilon}\,.$

Then, having fixed $\tilde{t}$, choose $n>m$ such that for the event $A_{n}$,
that in the time interval $(\tilde{t},1)$ at most $n$ hyperplanes of
$\hat{X}^{*}$ intersect $W$,

$\mathbb{P}(A_{n})=\sum_{i=0}^{n}\frac{((1-{\tilde{t}})\,\Lambda([W]))^{i}}{i!}{\rm
exp}\left(-(1-{\tilde{t}})\,\Lambda([W])\right)>1-{\varepsilon}\,.$

Note that $A\cap A_{n}\subseteq\\{W\subseteq\hat{z}_{n}\\}$ .

By the characteristic property of Poisson point processes, the restricted
point processes on disjoint sets, namely $\hat{X}^{*}\cap([W]\times(t^{*},1))$
and $\hat{X}^{*}\cap([W]^{c}\times(0,{\tilde{t}}))$, are independent. Hence,
for $B_{m}\in\bigvee_{k\leq m}{\cal F}_{k}$ and $C\in\bigvee_{k\geq n}{\cal
F}_{k}$ and with $D:=E\cap E_{m}\cap A\cap A_{n}$ we have that the events
$B_{m}$ and $C$ are conditionally independent under the condition $D$ and thus

$\displaystyle\left|\mathbb{P}(B_{m}\cap
C)-\mathbb{P}(B_{m})\mathbb{P}(C)\right|$ $\displaystyle=$
$\displaystyle\left|\mathbb{P}(B_{m}\cap C\cap D)+\mathbb{P}(B_{m}\cap C\cap
D^{c})-\mathbb{P}(B_{m})\mathbb{P}(C)\right|$ $\displaystyle\leq$
$\displaystyle\left|\mathbb{P}(B_{m}\cap D)\mathbb{P}(C\cap
D)/\mathbb{P}(D)-\mathbb{P}(B_{m})\mathbb{P}(C)\right|+\mathbb{P}(B_{m}\cap
C\cap D^{c})$ $\displaystyle<$
$\displaystyle\frac{8\,{\varepsilon}+16\,{\varepsilon}^{2}}{1-4{\varepsilon}}\,.$

Because $B\in{\cal F}_{\infty}\subseteq\bigvee_{k\geq n}{\cal F}_{k}$ we have
$\left|\mathbb{P}(B_{m}\cap
B)-\mathbb{P}(B_{m})\mathbb{P}(B)\right|<\frac{8\,{\varepsilon}+16\,{\varepsilon}^{2}}{1-4{\varepsilon}}\,$.
Then $\mathbb{P}(B\Delta B_{m})<{\varepsilon}$ yields $\left|\mathbb{P}(B\cap
B)-\mathbb{P}(B)\mathbb{P}(B)\right|<\frac{8\,{\varepsilon}+16\,{\varepsilon}^{2}}{1-4{\varepsilon}}+2\,{\varepsilon}$
for all ${\varepsilon}>0$, and hence $P(B)=0$ or 1. $\Box$

The tail triviality of the zero-cell process is now used to show further 0-1
probabilities.

###### Corollary 1

Let $\Lambda$ be a translation invariant measure on the space of hyperplanes
in $\mathbb{R}^{d}$, satisfying (1) and (2), and let $G:{{\cal
P}_{d}}\to[0,\infty)$ be a measurable functional on the set of $d$-dimensional
polytopes.

(a) Then

$\mathbb{P}\left(\exists m\in\mathbb{Z}:\ \sum_{i\leq
m}G\left(\tilde{z}^{o}_{(i)}\right)^{-1}<\infty\right)\in\\{0,\,1\\}\,.$ (7)

(b) Let $(\tau^{\prime}_{i},\,i\in\mathbb{Z})$ be a sequence of i.i.d. random
variables, independent of $\hat{X}^{*}$. Then

$\mathbb{P}\left(\exists m\in\mathbb{Z}:\ \sum_{i\leq
m}G\left(\tilde{z}^{o}_{(i)}\right)^{-1}\tau^{\prime}_{i}<\infty\right)\in\\{0,\,1\\}\,$
(8)

and

(c)

$\mathbb{P}\left(\exists m\in\mathbb{Z}:\ \sum_{i\leq
m}G\left(\tilde{z}^{o}_{(i)}\right)^{-1}\tau^{\prime}_{i}<\infty\right)=\mathbb{P}\left(\exists
m\in\mathbb{Z}:\ \sum_{i\leq
m}G\left(\tilde{z}^{o}_{(i)}\right)^{-1}<\infty\right)\,.$ (9)

Proof By the measurability of $G$, the event $\left\\{\exists m\in\mathbb{Z}:\
\sum_{i\leq m}G\left(\tilde{z}^{o}_{(i)}\right)^{-1}<\infty\right\\}$ is an
element of the tail-$\sigma$-algebra ${\cal F}_{\infty}$, and we can apply
Lemma 1.

If $(\tau^{\prime}_{i},\,i\in\mathbb{Z})$ be a sequence of i.i.d. random
variables which is independent of $\hat{X}^{*}$, then this result can be
extended straightforwardly to the product-$\sigma$-algebras ${\cal
F}_{n}\otimes{\cal F}_{n}^{\tau}$, $n\in\mathbb{N}_{0}$, where ${\cal
F}_{n}:=\sigma(\hat{z}_{n})$, $n\in\mathbb{N}_{0}$ and ${\cal
F}_{n}^{\tau}:=\sigma(\tau_{n})$. This yields (8).

To prove (9), first note that a.s. non-explosion of the process

$\left(\sum_{-j\leq i\leq
m}G\left(\tilde{z}^{o}_{(i)}\right)^{-1}\tau^{\prime}_{i},\,j\in\mathbb{N}\right)$

means that $\mathbb{P}\left(\sum_{i\leq
m}G\left(\tilde{z}^{o}_{(i)}\right)^{-1}\tau^{\prime}_{i}=\infty\right)=1$. It
is well-known in the theory of birth processes, see for example Proposition
13.5 of [7], that this is equivalent to $\mathbb{P}\left(\sum_{i\leq
m}G\left(\tilde{z}^{o}_{(i)}\right)^{-1}=\infty\right)=1$, which means that
the property of non-explosion only depends on the series of the expectations
$G\left(\tilde{z}^{o}_{(i)}\right)^{-1}$ of the holding times of the
respective states of the process. Hence, for the respective complements

$\mathbb{P}\left(\sum_{i\leq
m}G\left(\tilde{z}^{o}_{(i)}\right)^{-1}\tau^{\prime}_{i}<\infty\right)=0\
\Leftrightarrow\ \mathbb{P}\left(\sum_{i\leq
m}G\left(\tilde{z}^{o}_{(i)}\right)^{-1}<\infty\right)=0\,.$

Because by (7) and (8) both probabilities can have the values $0$ or $1$ only,
we obtain (9). $\Box$

In [15] the rescaled time-stationary process of zero-cells
$(a^{t}\tilde{z}^{o}_{a^{t}},t\in\mathbb{R})$ for $a>1$ was studied in more
detail and it was shown that this is a Bernoulli flow with infinite entropy.

##  4 Cell division processes

We consider a class of tessellation-valued Markov processes $(T_{t},t>0)$
which are characterized by the laws of the random life times of the cells and
the laws for the division of the cells by a random hyperplane at the end of
their life times. Furthermore, given the state of the process, conditional
independence of the future development in different cells is assumed. Denote
by $[z]:=\\{h\in{\cal H}:\,h\cap z\not=\emptyset\\}$ the set of all
hyperplanes that intersect the cell $z\in{{\cal P}_{d}}$.

###### Definition 2

Let be $\mathbb{Q}=(\mathbb{Q}_{[z]},\,z\in{{\cal P}_{d}})$, where
$\mathbb{Q}_{[z]}$ is a probability measure on the set $[z]$. And let
$G:{{\cal P}_{d}}\to[0,\infty)$ be a measurable functional defined on the set
of polytopes.

A random tessellation-valued process $(T_{t},t>0)$ is called an (L-$G$)
(D-$\mathbb{Q}$) cell division process if for all $t>0$ and all cells $z\in
T_{t}$:

  1. (L-G)

If $G(z)>0$, then the cell $z$ has a random life time which is exponentially
distributed with parameter $G(z)$ and, given $z$, conditionally independent of
$(T_{t^{\prime}},0<t^{\prime}<t)$ and of all other cells of $T_{t}$. If
$G(z)=0$ then the life time is infinite, which means that $z$ is never
divided.

  2. (D-$\mathbb{Q}$)

At the end of its life time, $z$ is divided by a random hyperplane $h_{z}$
with the law $\mathbb{Q}_{[z]}$, and also, given $z$, the hyperplane $h_{z}$
is conditionally independent of $(T_{t^{\prime}},0<t^{\prime}<t)$ and of all
other cells of $T_{t}$ and independent of the life time of $z$.

It is not trivial to see whether such a cell division process $(T_{t},t>0)$
exists for given $G$ and $\mathbb{Q}$, because there is no appropriate initial
tessellation at time $t=0$. Even if it is clear how to perform the cell
division dynamics in a bounded polytope $W$, a window, a consistent extension
to the whole space $\mathbb{R}^{d}$ is by no means straightforward.

Examples of interest for such rules can be related to translation invariant
hyperplane measures $\Lambda$ which satisfy (1) and (2).

We mean by

  1. (L-$\Lambda$)

…that the life time of each cell $z$ is exponentially distributed with
parameter $\Lambda([z])$,

  2. (L-$V_{d}$)

…that the life time of each cell $z$ is exponentially distributed with
parameter $V_{d}(z)$, the volume of $z$,

  3. (D-$\Lambda$)

…that the law of the random hyperplane dividing a cell $z$ is
$\mathbb{Q}_{[z]}=\Lambda([z])^{-1}\,\Lambda(\cdot\cap[z])$, which is the
probability measure on $[z]$, induced by $\Lambda$.

The STIT tessellation process driven by $\Lambda$ is a cell division process
with (L-$\Lambda$) and (D-$\Lambda$). In [18] it was shown that in the class
of cell division processes defined above, only the (L-$\Lambda$) (D-$\Lambda$)
model has the property of spatial consistency which is sufficient for its
existence. Therefore it is of interest to show the existence of further cell
division processes without requiring spatial consistency.

###  4.1 A sufficient condition for the existence of a cell division process
in $\mathbb{R}^{d}$

Let $({{\cal P}_{d}},{\cal B}({{\cal P}_{d}}))$ be the measurable space of
$d$-dimensional polytopes, where ${\cal B}({{\cal P}_{d}})$ is the Borel
$\sigma$-algebra with respect to the Fell topology on the space of closed
subsets of $\mathbb{R}^{d}$, see [24]. A functional $G:{{\cal
P}_{d}}\to[0,\infty)$ is called monotone if $G(z_{1})\leq G(z_{2})$ for all
$z_{1},z_{2}\in{{\cal P}_{d}}$ with $z_{1}\subseteq z_{2}$. It is called
invariant under translations if $G(z)=G(z+x)$ for all $z\in{{\cal P}_{d}}$ and
all $x\in\mathbb{R}^{d}$.

###### Theorem 1

Let $\Lambda$ be a translation invariant measure on the space of hyperplanes
in $\mathbb{R}^{d}$, satisfying (1) and (2), and let $G:{{\cal
P}_{d}}\to[0,\infty)$ be a measurable, monotone and translation invariant
functional on the set of $d$-dimensional polytopes. If for the random sequence
$\left(\tilde{z}^{o}_{(i)},\,i\in\mathbb{Z}\right)$ defined in (4) holds that
a.s.

$\exists m\in\mathbb{Z}:\quad\sum_{i\leq
m}G\left(\tilde{z}^{o}_{(i)}\right)^{-1}<\infty$ (10)

then there exists a Markov tessellation-valued process $(T_{t},t>0)$, which is
an (L-$G$) (D-$\Lambda$) cell division process according to Definition 2.

Proof The idea of the proof is inspired by the global construction of STIT
tessellations, described in [17]. The key is a backward construction for
$t\downarrow 0$ of the process $(z^{o}_{t},t>0)$ of zero-cells of
$(T_{t},t>0)$.

The auxiliary process $(\hat{X}_{t},t>0)$ of Poisson hyperplane tessellations
has the division law for its zero-cells which is exactly the (D-$\Lambda$)
rule. Hence the sizes and the shapes of the zero-cells of the Poisson
hyperplane tessellations can be used for the construction of
$(z^{o}_{t},t>0)$. What has to be changed is the ’clock’. The time axis has to
be transformed such that the pure jump process of zero cells receives jump
times according to (L-$G$), that is, the life times of the respective cells
are exponentially distributed, and the parameter is the functional value of
$G$ for these cells.

We use the random sequence $(\tilde{z}^{o}_{(i)},\,i\in\mathbb{Z})$ defined in
(4) for the construction, and we assign appropriate holding times to it.

Let $(\tau^{\prime}_{i},\,i\in\mathbb{Z})$ be a sequence of i.i.d. random
variables, exponentially distributed with parameter 1, and this sequence has
to be independent from $\hat{X}^{*}$. Define $(\tau_{i},\,i\in\mathbb{Z})$ as

$\tau_{i}:=\sum_{j\leq i-1}G(\tilde{z}^{o}_{(j)})^{-1}\,\tau^{\prime}_{j},\
i\in\mathbb{Z}.$ (11)

Now the process $(z^{o}_{t},t>0)$ is defined by

$z^{o}_{t}:=\tilde{z}^{o}_{(i)}\quad\mbox{ for }\tau_{i}\leq t<\tau_{i+1},\
i\in\mathbb{Z}.$ (12)

Thus, for the existence of the process $(z^{o}_{t},\,t>0)$ it is sufficient
that a.s. there is an $i\in\mathbb{Z}$ such that $\tau_{i}$ is finite. By (9)
it follows that (10) is equivalent to it. This implies also that
$\lim_{j\to-\infty}\tau_{j}=0$, a.s., and hence
$\bigcup_{t>0}z^{o}_{t}=\mathbb{R}^{d}$, by to (3).

Having the existence of the process $(z^{o}_{t},\,t>0)$ of zero-cells, the
process $(T_{t},\,t>0)$ is completed as follows. Denote by cl the topological
closure of a subset of $\mathbb{R}^{d}$. At each time $\tau_{i}$ , when the
process $(z^{o}_{t},\,t>0)$ has a jump from $\tilde{z}^{o}_{(i-1)}$ to
$\tilde{z}^{o}_{(i)}$, an (L-$G$) (D-$\Lambda$) cell division process
according to Definition 2 is launched in the new separated cell
$\textup{cl}\left(\tilde{z}^{o}_{(i-1)}\setminus\tilde{z}^{o}_{(i)}\right)$.

It remains to show that this actually yields a tessellation $T_{t}$ for all
$t>0$, and that the process $(T_{t},\,t>0)$ is an (L-$G$) (D-$\Lambda$) cell
division process.

Condition (b) of Definition 1 is obviously satisfied. To show (a), consider an
arbitrary point $x\in\mathbb{R}^{d}$. Then (3) implies that $x\in z^{o}_{s}$
for $0<s<t_{x}$, where $t_{x}$ is the time when $x$ and $o$ are separated by a
hyperplane. Thus the construction guarantees that for all $t>0$ any point
$x\in\mathbb{R}^{d}$ belongs to a the interior or to the boundary of a cell of
$T_{t}$.

To prove property (c) of Definition 1, consider a compact set
$C\subset\mathbb{R}^{d}$ and a time $t_{0}>0$ such that $C\subset
z^{o}_{t_{0}}$. By (3) such a time exists. Thus for all $t\leq t_{0}$, there
is exactly one cell in $T_{t}$ that has a nonempty intersection with $C$. For
$t>t_{0}$ the the number of cells inside $z^{o}_{t_{0}}$ of the cell division
process can be described by a birth process. By the monotonicity and
translation invariance of $G$, the birth rates are dominated by
$G(z^{o}_{t_{0}})\times\,number\ of\ cells\ in\ G(z^{o}_{t_{0}})$. Hence, at
any time $t>0$, the number of cells intersecting $C$ is a.s. finite.

Finally, to see that the process $(T_{t},\,t>0)$ is an (L-$G$) (D-$\Lambda$)
cell division process, observe that the division rule for the zero-cell
process is induced by the Poisson hyperplane process $\hat{X}$ and thus the
(D-$\Lambda$) rule is realized. The definition in (12) yields that the holding
time of the state $z^{o}_{(i)}$ is
$\tau_{i+1}-\tau_{i}=G(\tilde{z}^{o}_{(i)})^{-1}\,\tau^{\prime}_{i}$ which
shows that (L-$G$) is satisfied for the zero-cell process. For cells which are
already separated from the zero-cell, the (L-$G$) and (D-$\Lambda$) rules are
satisfied by the definition of the construction. $\Box$

In the rest of this paper, we mean by an (L-$G$) (D-$\Lambda$) process
$(T_{t},\,t>0)$ always the tessellation-valued process, defined in the proof.
Note that the proof does not imply that the distribution of (L-$G$)
(D-$\Lambda$) process is uniquely determined by $G$ and $\Lambda$.

An example where the sufficient condition (10) is obviously not satisfied, is
$G(z)=c$ for all $z\in{{\cal P}_{d}}$ and some fixed $c>0$. It is not known
whether a cell division tessellation process exists for this $G$, but the
simulations in [3] for the corresponding ’equally likely’ case suggest that
the cell division does not yield a tessellation according to Definition 1,
because the local finiteness condition (c) seems to be violated.

###  4.2 Stationarity in space

Now it will be shown that the translation invariance of the hyperplane measure
$\Lambda$ and of the functional $G$ are sufficient for the spatial
stationarity of the constructed tessellations at any fixed time $t>0$.

###### Theorem 2

If the assumptions of Theorem 1 are satisfied, then for all $t>0$ the random
tessellation $T_{t}$, described in the proof of the theorem, is a spatially
stationary (or homogeneous) tessellation.

Proof The proof can be sketched as follows. Because the hyperplane measure
$\Lambda$ defined in (1) is invariant under translations of $\mathbb{R}^{d}$,
the distribution of the auxiliary Poisson process $\hat{X}^{*}$ is invariant
under translations by $(x,0)\in\mathbb{R}^{d}\times\\{0\\}$. Thus, denoting by
$\left(\tilde{z}^{o}_{(i)}(\hat{X}^{*}),\,i\in\mathbb{Z}\right)$ the process
of zero-cells generated by $\hat{X}^{*}$,

$\left(\tilde{z}^{o}_{(i)}(\hat{X}^{*}+(x,0)),\,i\in\mathbb{Z}\right)\stackrel{{\scriptstyle
D}}{{=}}\left(\tilde{z}^{o}_{(i)}(\hat{X}^{*}),\,i\in\mathbb{Z}\right)\quad\mbox{
for all }x\in\mathbb{R}^{d}.$

This means the invariance under translations of $\mathbb{R}^{d}$ of the
distribution of the process $(\tilde{z}^{o}_{(i)},\,i\in\mathbb{Z})$ defined
in (4).

Furthermore, the translation invariance of $G$ guarantees that the life time
distributions of the cells remain translation invariant too. Finally, because
$\Lambda$ is invariant under translations, it follows that
$\Lambda([z+x])^{-1}\,\Lambda(A+x\cap[z+x])=\Lambda([z])^{-1}\,\Lambda(A\cap[z])$
for all $z\in{{\cal P}_{d}}$, $x\in\mathbb{R}^{d}$ and Borel sets
$A\subset[z]$. Hence, the division law is translation-equivariant. $\Box$

###  4.3 Construction of the cut-out appearing in a bounded window $W$

For simulations of the model it is of interest to construct the restriction
$(T_{t}\land W,t>0)$ of $(T_{t},\,t>0)$ to a bounded window $W\in{{\cal
P}_{d}}$, where $T_{t}\land W:=\\{z\cap W:z\in T_{t},\textup{int}(z\cap
W)\not=\emptyset\\}$. In [20] it was shown, that for STIT tessellations the
(L-$\Lambda$) (D-$\Lambda$) cell division process can be launched in $W$
without regarding boundary effects, because the STIT model is spatially
consistent. And in [18] it was shown that all the other models considered in
the present paper are not spatially consistent which means that for the
construction of $(T_{t}\land W,t>0)$ also information outside of $W$ is
needed.

The construction described in the proof of Theorem 1 also shows how
$(T_{t}\land W,t>0)$ can be realized. For a given window $W$ construct a zero-
cell $\tilde{z}^{o}_{(i_{W})}$ from the process
$(\tilde{z}^{o}_{(i)},\,i\in\mathbb{Z})$ defined in (4) such that
$W\subset\tilde{z}^{o}_{(i_{W})}$. This can be done as follows. Let be $R>0$
such that $W\subset{\sf B}_{R}$. Simulate the restricted Poisson point process
$\hat{X}^{*}\cap([{\sf B}_{nR}]\times(0,\frac{1}{n}))$, starting with $n=2$.
If the generated zero-cell contains $W$, then choose it as
$\tilde{z}^{o}_{(i_{W})}$; otherwise update $n:=n+1$ until an appropriate
zero-cell is simulated.

Then launch the (L-$G$) (D-$\Lambda$) cell division process inside
$\tilde{z}^{o}_{(i_{W})}$. Its restriction to $W$ has exactly the distribution
of $(T_{t+\tau_{i_{W}}}\land W,t>0)$. Here, the jump time $\tau_{i_{W}}$ of
the zero-cell process is unknown because we do not know the value of the
series (11).

##  5 The Mondrian directional distribution

Now we consider a particular class of directional distributions $\varphi$ and
functionals $G$ where the sufficient condition of Theorem 1 is satisfied.

Denote by $e_{1},\ldots,e_{d}$ the orthonormal base of $\mathbb{R}^{d}$ with
$e_{k}:=(0,\ldots,0,1,0\ldots 0)$ where the 1 is at the $k$-th position,
$k\in\\{1,\ldots,d\\}$. Let $\delta_{e_{k}}$ denote the Dirac probability
measure defined on $S^{d-1}$ and with mass 1 on $e_{k}$.

In this section we consider spherical directional distributions of the form

$\varphi=\frac{1}{2}\sum_{k=1}^{d}p_{k}(\delta_{e_{k}}+\delta_{-e_{k}}),\quad
p_{k}>0,\ \sum_{k=1}^{d}p_{k}=1.$ (13)

Referring to recent papers like [23], we will call them Mondrian directional
distributions.

The random tessellations generated by Mondrian directional distributions
consist of cuboids. The following Theorems 3 and 4 provide first concrete
models of cell-division processes in $\mathbb{R}^{d}$ that differ from the
STIT model.

Figure 1: Simulations of cell division tessellations in a quadratic window
with (L-$\Lambda$) (left panel) and (L-$V_{2}$) (right panel), both with
(D-$\Lambda$). The directional distribution is
$\varphi=\frac{1}{2}\delta_{e_{1}}+\frac{1}{2}\delta_{e_{2}}$. (Generated with
the software [9].)

###  5.1 Existence results

For a $d$-dimensional cuboid $z$ with all $(d-1)$-dimensional faces parallel
or orthogonal to the axes in $\mathbb{R}^{d}$, denote by by
$l^{(1)},\ldots,l^{(d)}$ the side lengths of the cuboid, where $l^{(k)}$ is
the length of the sides that are parallel to $e_{k}$, $k\in\\{1,\ldots,d\\}$.
Denote $S(z):=\sum_{k=1}^{d}l^{(k)}$ if $z$ is a cuboid and $S(z)=1$
elsewhere.

###### Theorem 3

Let $\Lambda$ be a translation invariant measure on the space of hyperplanes
in $\mathbb{R}^{d}$, satisfying (1) where $\varphi$ is a Mondrian directional
distribution (13). Then there exists a Markov tessellation-valued process
$(T_{t},t>0)$, which is an (L-$S$) (D-$\Lambda$) cell division process
according to Definition 2.

Proof Consider the auxiliary Poisson process $\hat{X}^{*}$ of hyperplanes that
are marked with birth times, that is the Poisson point process on ${\cal
H}\times(0,\infty)$ with the intensity measure $\Lambda\otimes\lambda_{+}$,
see Section 2, and $\Lambda$ defined in (1) with $\varphi$ as in (13). We show
that the sufficient condition of Theorem 1 is satisfied for the functional
$G=S$, which is monotone and translation invariant on the class of cuboids.

The Poisson point process $\hat{X}^{*}$ can be described as a superposition of
$d$ i.i.d. Poisson point processes $\hat{X}^{(k)*}$, that is
$\hat{X}^{*}\stackrel{{\scriptstyle D}}{{=}}\bigcup_{k=1}^{d}\hat{X}^{(k)*}$,
and $\hat{X}^{(k)*}$ is a Poisson point process on ${\cal H}\times(0,\infty)$
with the intensity measure $p_{k}\,\Lambda^{(k)}\otimes\lambda_{+}$, and
$\Lambda^{(k)}$ as defined in (1) with
$\varphi^{(k)}=\frac{1}{2}(\delta_{e_{k}}+\delta_{-e_{k}})$,
$k\in\\{1,\ldots,d\\}$ (for which condition (2) is obviously not satisfied).

For $k\in\\{1,\ldots,d\\}$ we define the Poisson point processes $\Phi^{k+}$
and $\Phi^{k-}$ on $(0,\infty)\times(0,\infty)$ as the birth time marked
processes of intersection points of the hyperplanes of $\hat{X}^{(k)*}$ with
the positive axis and the negative axis in direction $e_{k}$ and $-e_{k}$,
respectively. Formally,

$\Phi^{k+}:=\\{(x,t)\in(0,\infty)\times(0,\infty):(h(e_{k},x),t)\in\hat{X}^{(k)*}\\}$

and

$\Phi^{k-}:=\\{(x,t)\in(0,\infty)\times(0,\infty):(h(-e_{k},x),t)\in\hat{X}^{(k)*}\\}$.

Both point processes have the intensity measure
$p_{k}\,\lambda_{+}\otimes\lambda_{+}$. Now we define the Markov chains

$M^{k+}=((x^{k+}_{n},t^{k+}_{n}),\,n\in\mathbb{N}_{0})$

with $(x^{k+}_{n},t^{k+}_{n})\in\Phi^{k+}$ and

$x^{k+}_{0}:=\min\\{x:\,(x,t)\in\Phi^{k+},t\leq 1\\}$,

$x^{k+}_{n+1}:=\min\\{x:\,(x,t)\in\Phi^{k+},t\leq t^{k+}_{n}\\}$,
$n\in\mathbb{N}_{0}$.

The times $t^{k+}_{n}$ are a.s. uniquely defined by the condition
$(x^{k+}_{n},t^{k+}_{n})\in\Phi^{k+}$. Correspondingly, $M^{k-}$ is defined,
replacing $k+$ by $k-$.

The properties of the Poisson process $\hat{X}^{*}$ yield that for fixed
$k\in\\{1,\ldots,d\\}$ the Markov chains $M^{k+}$ and $M^{k-}$ are i.i.d.

Let us consider some properties of $M^{k+}$. The random variable $x^{k+}_{0}$
is exponentially distributed with parameter $p_{k}$, and $t^{k+}_{0}$ is
uniformly distributed in the interval $(0,1)$. Then, $x^{k+}_{n+1}-x^{k+}_{n}$
is exponentially distributed with the parameter $p_{k}\,t^{k+}_{n}$ and
independent of $x^{k+}_{0},\ldots,x^{k+}_{n}$ and, given $t^{k+}_{n}$,
conditionally independent of $t^{k+}_{0},\ldots,t^{k+}_{n-1}$. Furthermore,
$t^{k+}_{n+1}$ is uniformly distributed in the interval $(0,t^{k+}_{n})$ and,
given $t^{k+}_{n}$, conditionally independent of
$x^{k+}_{0},\ldots,x^{k+}_{n}$ and of $t^{k+}_{0},\ldots,t^{k+}_{n-1}$.

Now consider the Markov chain $((S_{i},t_{i}),\,i\in\mathbb{Z}_{-})$ where
$(t_{i},\,i\in\mathbb{Z})$ is the ordered sequence of jump times of
$(\tilde{z}^{o}_{t},t>0)$ with $t_{0}<1<t_{1}$, as defined in Section 2, and
$S_{i}:=S\left(\tilde{z}^{o}_{(i)}\right)$. Thus we have
$t_{0}=\max\\{t^{k+}_{0},t^{k-}_{0},k\in\\{1,\ldots,d\\}\\}$, and

$S_{0}=\sum_{k=1}^{d}x^{k+}_{0}+x^{k-}_{0}.$

By recursion,
$t_{i-1}=\max\\{t^{k+}_{n}<t_{i},t^{k-}_{n}<t_{i},k\in\\{1,\ldots,d\\},\
n\in\mathbb{Z}_{-}\\},$ and

$S_{i-1}=\begin{cases}S_{i}+x^{k+}_{m+1}-x^{k+}_{m}&\textup{ if
}t_{i}=t^{k+}_{m}\ \textup{ for some }m\in\mathbb{Z}_{-}\ ,\\\
S_{i}+x^{k-}_{m+1}-x^{k-}_{m}&\textup{ if }t_{i}=t^{k-}_{m}\ \textup{ for some
}m\in\mathbb{Z}_{-}\ .\end{cases}$ (14)

Let $(R_{i},\,i\in\mathbb{Z})$ be a sequence of i.i.d. random variables with
the probability density $f(r)=2dr^{2d-1}{\bf 1}\\{0<r<1\\}$, that is, $R_{i}$
has the distribution of the maximum of $2d$ i.i.d. random variables that are
uniformly distributed in the interval $(0,1)$.

Thus the law of the time component $t_{i}$ in the Markov chain
$((S_{i},t_{i}),\,i\in\mathbb{Z}_{-})$ can be described by

$t_{0}\stackrel{{\scriptstyle D}}{{=}}R_{0}\,,$

and by recursion

$t_{i-1}\stackrel{{\scriptstyle D}}{{=}}t_{i}\,R_{i-1}\,,\quad
i\in\mathbb{Z}_{-}\,.$

Thus

$(t_{i},\,\,i\in\mathbb{Z}_{-})\stackrel{{\scriptstyle
D}}{{=}}\textstyle{\left(\prod_{j=i}^{0}R_{j},\,\,i\in\mathbb{Z}_{-}\right)}\,.$

The law of $S_{i-1}-S_{i}$ depends on both $t_{i}$ and the index $k^{+}$ or
$k^{-}$. It is the exponential distribution with the parameter $p_{k}\,t_{i}$,
not depending on the sign of the index $k$.

Let $(\kappa_{i},\,i\in\mathbb{Z})$ be a sequence of i.i.d. random variables,
exponentially distributed with parameter 1. This sequence is assumed to be
independent of all the other random variables we considered so far. Then the
conditional distribution is

$\displaystyle S_{i-1}-S_{i}\stackrel{{\scriptstyle
D}}{{=}}\left(p_{k}\,\prod_{j=i}^{0}R_{j}\right)^{-1}\kappa_{i}\leq
p_{\min}^{-1}\left(\prod_{j=i}^{0}R_{j}\right)^{-1}\kappa_{i},$ (15)
$\displaystyle\textup{under the condition that }t_{i}=t^{k+}_{m}\textup{ or
}t_{i}=t^{k-}_{m}\textup{ for some }m\in\mathbb{N}_{0}\,,$

with $p_{\min}:=\min\\{p_{1},\ldots,p_{k}\\}$.

Thus a sufficient condition for

$\sum_{i\leq 0}S_{i}^{-1}=S_{0}^{-1}+\sum_{i\leq
0}\left(S_{0}+\sum_{\ell=i}^{0}(S_{\ell-1}-S_{\ell})\right)^{-1}<\infty\quad
a.s.$

is

$\sum_{i\leq
0}\left(\sum_{\ell=i}^{0}\left(\prod_{j=\ell}^{0}R_{j}\right)^{-1}\kappa_{\ell}\right)^{-1}<\infty\quad
a.s.$ (16)

Note that in (16) the item $S_{0}^{-1}$ is neglected in order to simplify the
technicalities. To verify (16) we provide an upper bound for the expectation
of the random variable on the left-hand side.

$\displaystyle\mathbb{E}\sum_{i\leq
0}\left(\sum_{\ell=i}^{0}\left(\prod_{j=\ell}^{0}R_{j}\right)^{-1}\kappa_{\ell}\right)^{-1}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle\\!\leq$ $\displaystyle\ \mathbb{E}\sum_{i\leq
0}\left(\sum_{\ell=i}^{i+1}\left(\prod_{j=\ell}^{0}R_{j}\right)^{-1}\kappa_{\ell}\right)^{-1}$
$\displaystyle=$ $\displaystyle\,\mathbb{E}\sum_{i\leq
0}\frac{\prod_{j=i+1}^{0}R_{j}}{R_{i}^{-1}\,\kappa_{i}+\kappa_{i+1}}$
$\displaystyle\leq$ $\displaystyle\,\mathbb{E}\sum_{i\leq
0}\frac{\prod_{j=i+1}^{0}R_{j}}{\kappa_{i}+\kappa_{i+1}}$ $\displaystyle=$
$\displaystyle\,\mathbb{E}\frac{1}{\kappa_{0}+\kappa_{1}}\ \sum_{i\leq
0}\left(\prod_{j=i+1}^{0}\mathbb{E}R_{j}\right)\,$ $\displaystyle=$
$\displaystyle\,\sum_{i\leq
0}\left(\prod_{j=i+1}^{0}\frac{2d}{2d+1}\right)<\infty\,.$

In the last equation we used that the sum of two i.i.d. exponentially
distributed random variables with parameter 1, has a gamma-distribution with
parameter $(2,1)$, also referred to as an Erlang distribution, and hence
$\mathbb{E}(\kappa_{0}+\kappa_{1})^{-1}=1$. Thus it is shown that (16) is
satisfied. $\Box$

In order to generalize the result for Mondrian directional distributions, we
consider the individual side lengths of the zero-cells. Denote by
$l_{i}^{(1)},\ldots,l_{i}^{(d)}$ the side lengths of the cuboid
$\tilde{z}^{o}_{(i)}$, where $l_{i}^{(k)}$ is the length of the sides that are
parallel to $e_{k}$, $k\in\\{1,\ldots,d\\}$, $i\in\mathbb{Z}$. The following
technical lemma prepares the proof of Theorem 4.

###### Lemma 2

With the assumptions of Therorem 3,

$\mathbb{P}\left(\exists i_{0}\in\mathbb{Z}\,\forall i\leq i_{0}\,\forall
k\in\\{1,\ldots,d\\}:\,l_{i}^{(k)}>d\right)=1.$

Proof Fix some $k\in\\{1,\ldots,d\\}$ and, using the notation in the proof of
Theorem 3, consider (15) which expresses the growth of the length
$l_{i}^{(k)}$ if $t_{i}=t^{k+}_{m}\textup{ or }t_{i}=t^{k-}_{m}\textup{ for
some }m\in\mathbb{N}_{0}$. The index $k$ is chosen with probability $p_{k}$.
Obviously,
$\left(p_{k}\,\prod_{j=i}^{0}R_{j}\right)^{-1}>1$. Hence, the events

$B_{i}:=\\{\kappa_{i}>d,\,t_{i}=t^{k+}_{m}\textup{ or
}t_{i}=t^{k-}_{m}\textup{ for some }m\in\mathbb{N}_{0}\\},\quad
i\in\mathbb{Z},$

which are independent, have a strictly positive probability that do not depend
on $i$. Thus the Borel-Cantelli Lemma yields that
$\mathbb{P}\left(\bigcap_{n\leq 0}\bigcup_{i\leq n}B_{i}\right)=1,$ that is
there occur infinitely many of the $B_{i}$ a.s. This, together with the
monotonicity of the sequence $(l_{i}^{(k)},\,i\in\mathbb{Z})$ yields that for
all $k\in\\{1,\ldots,d\\}$ we have

$\mathbb{P}\left(\exists i_{0}(k)\in\mathbb{Z}\,\forall i\leq
i_{0}(k):\,l_{i}^{(k)}>d\right)=1,$

and hence

$\mathbb{P}\left(\forall k\in\\{1,\ldots,d\\}\,\exists
i_{0}(k)\in\mathbb{Z}\,\forall i\leq i_{0}(k):\,l_{i}^{(k)}>d\right)=1.$

Because there are only finitely many $k$, the assertion of the lemma follows.
$\Box$

Lemma 2 can be used to check the sufficient condition (10) when the functional
$G$ is chosen as a function of the side lengths of the zero cell. Here we
consider the important cases of the intrinsic volumes $V_{n}$,
$n\in\\{1,\ldots,d\\}$. The functional $V_{d}$ is the volume, $V_{1}$ the mean
width (or breadth) up to a constant factor. For $n=3$ the intrinsic volume
$V_{2}$ is, up to a constant factor, the surface area. The intrinsic volume
$V_{0}$ is the Euler characteristic which has the value 1 for all nonempty
convex sets, and hence it is clear that for $G=V_{0}$ the sufficient condition
(10) is not satisfied.

A definition of the intrinsic volumes for convex bodies can be found in [24].
Here we make use of a formula for cuboids which is a particular case of the
formula for polytopes, see (14.35) of [24]. For $n\geq 1$

$V_{n}(\tilde{z}^{o}_{(i)})=\sum_{1\leq k_{1}<\ldots<k_{n}\leq d}\ \
\prod_{j=1}^{n}l_{i}^{(k_{j})}\,.$ (17)

###### Theorem 4

Let $\Lambda$ be a translation invariant measure on the space of hyperplanes
in $\mathbb{R}^{d}$, satisfying (1) where $\varphi$ is a Mondrian directional
distribution (13). Then, for all $n\in\\{1,\ldots,d\\}$, there exists a Markov
tessellation-valued process $(T_{t},t>0)$, which is an (L-$V_{n}$)
(D-$\Lambda$) cell division process according to Definition 2.

Proof It is sufficient to show that (10) is satisfied for $G=V_{n}$ for all
$n\in\\{1,\ldots,d\\}$. For $n=1$ it follows immediately from Theorem 3. For
$n\in\\{2,\ldots,k\\}$ and $i\in\mathbb{Z}$,

$\left(\sum_{k_{1}<\ldots<k_{n}}\,\prod_{j=1}^{n}l_{i}^{(k_{j})}\right)-\sum_{k=1}^{d}l_{i}^{(k)}=\frac{1}{{d-1\choose
n-1}}\sum_{k=1}^{d}l_{i}^{(k)}\left(\sum_{{k_{1}<\ldots<k_{n-1}}\atop{k_{j}\not=k}}\left(\frac{{d\choose
n}}{d}\prod_{j=1}^{n-1}l_{i}^{(k_{j})}-1\right)\right).$

Note that ${d\choose d}/d=1/d$ and ${d\choose n}/d\geq 1$ for
$n\in\\{1,\ldots,d-1\\}$. By Lemma 2 there exists a.s. an $i_{0}\in\mathbb{Z}$
such that for all $i\leq i_{0}$ and all $k\in\\{1,\ldots,d\\}$ the side
lengths $l_{i}^{(k)}>d$, which implies that the difference on the left-hand
side is positive. Recall that $S_{i}=\sum_{k=1}^{d}l_{i}^{(k)}$ for all
$i\in\mathbb{Z}$, and that in the proof of of Theorem 3 it was shown that
$\sum_{i\leq 0}S_{i}^{-1}<\infty\quad a.s.$ Hence, for $n\in\\{2,\ldots,k\\}$
also

$\sum_{i\leq
0}\left(\sum_{k_{1}<\ldots<k_{n}}\,\prod_{j=1}^{n}l_{i}^{(k_{j})}\right)^{-1}<\infty\quad
a.s.$

and by (17),

$\sum_{i\leq 0}V_{n}\left(\tilde{z}^{o}_{(i)}\right)^{-1}\,<\infty\quad a.s.$

$\Box$

###### Remark 1

Theorem 4 can easily be generalized to (L-$V_{n}^{\alpha}$) (D-$\Lambda$) cell
division process, for exponents $\alpha\geq 1$. The larger $\alpha$, the
longer will be the life time of cells $z$ with $V_{n}(z)<1$ and the shorter
will be the life time if $V_{n}(z)>1$.

Note that for the Mondrian distribution defined in (13) with
$p_{1}=\ldots=p_{d}=1/d$, the (L-$V_{1}$) (D-$\Lambda$) cell division process
is – up to a scaling factor – the same as the STIT tessellation process driven
by $\Lambda$, which is a (L-$\Lambda$) (D-$\Lambda$) cell division process.

###  5.2  Distribution of cell volumes

Now we consider the particular cell division process in $\mathbb{R}^{d}$ with
volume-weighted life time distribution (L-$V_{d}$) and (D-$\Lambda$), with a
Mondrian directional distribution (13). We make use of an idea used by Cowan
in [3] developed for so-called ’geometry-independent apportionment of volume’
(GIA) which means a division rule such that the ratio of the volume of a
daughter cell and the volume of the mother cell is uniformly distributed on
the interval $(0,1)$.

In contrast to this, in Subsection 5.3 we will consider an (L-$V_{d}$)
(D-$\Lambda$) process in a bounded window $W$ and describe its relation to
fragmentation.

For a random stationary tessellation of $\mathbb{R}^{d}$ the concept of the
’typical cell’ is formally defined using the Palm measure, see p. 450 of [24]
for example. Intuitively, it can be imagined as a ’cell, cosen at random,
where all cells have the same chance to be chosen’.

###### Lemma 3

Let $\Lambda$ be a translation invariant measure on the space of hyperplanes
in $\mathbb{R}^{d}$, satisfying (1) where $\varphi$ is a Mondrian directional
distribution (13), and let $(T_{t},\,t>0)$ be the (L-$V_{d}$) (D-$\Lambda$)
cell division process as described in the proof of Theorem 1. Then for all
$t>0$ the law of the volume $V_{d}(z)$ of the typical cell of $T_{t}$ is the
exponential distribution with parameter $t$, and the law of the volume
$V_{d}(z^{o}_{t})$ of the zero-cell of $T_{t}$ is the gamma-distribution with
parameter $(2,t)$, which is also referred to as an Erlang distribution.

Proof The volumes of cells of a tessellation $T_{t}$, $t>0$, are represented
by the lengths of intervals on $\mathbb{R}$. The idea of the proof is the
following: The tessellation $T_{t}$ is mapped to a point process
$(y^{(j)},\,j\in\mathbb{Z})$ on $\mathbb{R}$ such that the lengths of the
intervals between adjacent points correspond to the volumes of the cells of
$T_{t}$.

In order to formalize this idea, it is an involved technical issue to bring
these intervals into an appropriate linear order. As before, we start with the
process of zero cells as defined in (12) and we use a method analogous to that
one in the proof of Theorem 3. But here we consider the differences of volumes
when cells are split.

If for some ${\varepsilon}>0$ the tessellation $T_{\varepsilon}$ is mapped to
a point process $(y_{\varepsilon}^{(j)},\,j\in\mathbb{Z})$ on $\mathbb{R}$
such that the lengths of the intervals between adjacent points correspond to
the volumes of the cells of $T_{\varepsilon}$, then the division of the cell
volumes in the continuation of the cell division process corresponds to a
Poisson point process that is superposed to
$(y_{\varepsilon}^{(j)},\,j\in\mathbb{Z})$. Because this can be done for any
${\varepsilon}>0$, one can conclude that the cell volumes of $T_{t}$ are
represented by the lengths of the interval between two adjacent points of a
Poisson point process. Summarizing the idea, we will show that for a Poisson
point process on $\mathbb{R}^{d}\times(0,\infty)$ with intensity maesure
$\lambda\otimes\lambda_{+}$

Formalizing this idea, it is an issue to bring these intervals into an
appropriate linear order. Let us begin with the process of zero cells as
defined in (12). We use a method analogous to that one in the proof of Theorem
3 above, but here we consider the differences of volumes when cells are split.

Fix an ${\varepsilon}>0$ and $i\in\mathbb{Z}$ such that
$\tau_{i}\leq{\varepsilon}<\tau_{i+1}$, where $\tau_{i}$ is defined in (11)
with $G=V_{d}$. Let
$\hat{y}_{\varepsilon}^{(-1)}<0<\hat{y}_{\varepsilon}^{(0)}$ be real valued
such that
$\hat{y}_{\varepsilon}^{(0)}-\hat{y}_{\varepsilon}^{(-1)}=V_{d}(z_{\tau_{i}})$,
and $0$ is uniformly distributed in the interval
$(\hat{y}_{\varepsilon}^{(-1)},\hat{y}_{\varepsilon}^{(0)})$

Initialize a counter $r:=0$ and a counter $\ell:=1$. If in (14)
$t_{i}=t^{k+}_{m}$ for some $k$ and $m$, then put
$\hat{y}_{\varepsilon}^{(1)}:=\hat{y}_{\varepsilon}^{(0)}+V_{d}(z_{\tau_{i-1}}\setminus
z_{\tau_{i}})$, $t^{(0)}:=\tau_{i}$, and update $r:=1$. If $t_{i}=t^{k-}_{m}$
for some $k$ and $m$, then define
$\hat{y}_{\varepsilon}^{(-2)}:=\hat{y}_{\varepsilon}^{(-1)}-V_{d}(z_{\tau_{i-1}}\setminus
z_{\tau_{i}})$, $t^{(-1)}:=\tau_{i}$, and update $\ell:=2$.

Having defined
$(\hat{y}_{\varepsilon}^{(-\ell)},t^{(-\ell)}),\ldots,(\hat{y}_{\varepsilon}^{(r)},t^{(r)})$
for some $\ell\in\mathbb{N}_{0}$ and $r\in\mathbb{N}_{0}$, the next item of
the sequence is: If in (14) $t_{i-r-\ell-1}=t^{k+}_{m}$ for some $k$ and $m$,
then
$\hat{y}_{\varepsilon}^{(r+1)}:=\hat{y}_{\varepsilon}^{(r)}+V_{d}(z_{\tau_{i-r-\ell-2}}\setminus
z_{\tau_{i-r-\ell-1}})$, $t^{(r+1)}:=\tau_{i-r-\ell-1}$, and update $r:=r+1$.
If $t_{i-r-\ell-1}=t^{k-}_{m}$ for some $k$ and $m$, then
$\hat{y}_{\varepsilon}^{(-\ell-1))}:=\hat{y}_{\varepsilon}^{(-\ell)}-V_{d}(z_{\tau_{i-r-\ell-2}}\setminus
z_{\tau_{i-r-\ell-1}})$, $t^{(-\ell-1)}:=\tau_{i-r-\ell-1}$, and update
$\ell:=\ell+1$.

This yields the sequence
$((\hat{y}_{\varepsilon}^{(j)},t^{(j)}),\,j\in\mathbb{Z})$ pertaining to the
process of zero cells. It has to be complemented by the points which
correspond to the divisions of the cells
$\textup{cl}(z_{\tau_{i-r-\ell-2}}\setminus z_{\tau_{i-r-\ell-1}})$ in the
time interval $(\tau_{i-r-\ell-1},t)$.

This can be described as follows. The volume of the cell
$\textup{cl}(z_{\tau_{i-r-\ell-2}}\setminus z_{\tau_{i-r-\ell-1}})$ equals the
length of the interval
$(\hat{y}_{\varepsilon}^{(r)},\hat{y}_{\varepsilon}^{(r+1)})$ or
$(\hat{y}_{\varepsilon}^{(-\ell-1)},\hat{y}_{\varepsilon}^{(-\ell)})$,
respectively. For the sake of simplicity, in the following we consider only
the part of $((\hat{y}_{\varepsilon}^{(j)},t^{(j)}),\,j\in\mathbb{Z})$ which
belongs to $(0,\infty)\times(0,\infty)$. The other part on
$(-\infty,0)\times(0,\infty)$ can be dealt with quite analogously. When the
cell $\textup{cl}(z_{\tau_{i-r-\ell-2}}\setminus z_{\tau_{i-r-\ell-1}})$ is
divided by a hyperplane, the respective interval is divided by a point such
that the two new interval lengths equal the two volumes of the daughter cells.
We define the new interval which is closer to $0\in\mathbb{R}$ to pertain to
the daughter cell which is contained in the half space of the dividing
hyperplane that contains the origin $o\in\mathbb{R}^{d}$. The point dividing
the interval is marked with the birth time of the hyperplane dividing the
cell. Thus the interval
$(\hat{y}_{\varepsilon}^{(r)},\hat{y}_{\varepsilon}^{(r+1)})$ is divided after
a life time, which is exponentially distributed with the parameter
$\hat{y}_{\varepsilon}^{(r+1)}-\hat{y}_{\varepsilon}^{(r)}$, and the dividing
point is – due to the translation invariance of $\Lambda$ – uniformly
distributed in this interval. The new daughter intervals are subsequently
divided, following analogous rules. Thus the birth time marked point process
appearing in the interval
$(\hat{y}_{\varepsilon}^{(r)},\hat{y}_{\varepsilon}^{(r+1)})$ has the same
distribution as the Poisson point process on
$(\hat{y}_{\varepsilon}^{(r)},\hat{y}_{\varepsilon}^{(r+1)})\times(\tau_{i-r-\ell-1},t)$
with the two-dimensional Lebesgue measure (restricted to this set) as its
intensity measure.

Taking all the marked points together, we obtain a point process
$Y_{\varepsilon}:=((y_{\varepsilon}^{(j)},t^{(j)}),\,j\in\mathbb{Z})$.

Given the point process
$Y_{\varepsilon}=((y_{\varepsilon}^{(j)},t^{(j)}),\,j\in\mathbb{Z})$ for some
time ${\varepsilon}>0$, and regarding the independence assumptions in
Definition 2 the generation of the point process
$Y_{t}=((y_{t}^{(j)},t^{(j)}),\,j\in\mathbb{Z})$ for $t>{\varepsilon}$ can be
described using an auxiliary Poisson point process $\Phi^{*}$ on
$\mathbb{R}\times(0,\infty)$ with intensity measure
$\lambda\otimes\lambda_{+}$.

For all $0<{\varepsilon}<t$ let be
$\Phi^{*}_{({\varepsilon},t)}:=\\{(x,t^{\prime})\in\Phi^{*}:\,{\varepsilon}<t^{\prime}<t\\}$.
We obtain that

$Y_{t}\stackrel{{\scriptstyle
D}}{{=}}Y_{\varepsilon}\cup\Phi^{*}_{({\varepsilon},t)},$ (18)

which is the superposition of two independent point processes.

The sequence $((\hat{y}_{\varepsilon}^{(j)},t^{(j)}),\,j\in\mathbb{Z})$
pertaining to the process of zero cells is monotone in ${\varepsilon}$ in the
following sense. For ${\varepsilon}_{2}<{\varepsilon}_{1}$ the sequence
$((\hat{y}_{{\varepsilon}_{2}}^{(j)},t^{(j)}),\,j\in\mathbb{Z})$ emerges from
$((\hat{y}_{{\varepsilon}_{1}}^{(i)},t^{(i)}),\,i\in\mathbb{Z})$ by deleting
the points with

$t^{(i+1)}>{\varepsilon}_{2}\mbox{ for }i\geq 0,\qquad\mbox{or}\qquad
t^{(i-1)}>{\varepsilon}_{2}\mbox{ for }i<0,$

and after this, a respective rearrangement of the indexes $i\in\mathbb{Z}$.
Accordingly,
$Y_{{\varepsilon}_{2}}=((y_{{\varepsilon}_{2}}^{(j)},t^{(j)}),\,j\in\mathbb{Z})$
can – up to a rearrangement of the indexes – be considered as a subsequence of
$Y_{{\varepsilon}_{1}}=((y_{{\varepsilon}_{1}}^{(j)},t^{(j)}),\,j\in\mathbb{Z})$.

Denoting by $Y_{({\varepsilon}_{2},{\varepsilon}_{1})}$ the difference between
$Y_{{\varepsilon}_{1}}$ and $Y_{{\varepsilon}_{2}}$ we can write

$Y_{{\varepsilon}_{2}}\cup\Phi^{*}_{({\varepsilon}_{2},t)}\stackrel{{\scriptstyle
D}}{{=}}Y_{{\varepsilon}_{2}}\cup\Phi^{*}_{({\varepsilon}_{2},{\varepsilon}_{1})}\cup\Phi^{*}_{({\varepsilon}_{1},t)}\stackrel{{\scriptstyle
D}}{{=}}Y_{{\varepsilon}_{2}}\cup
Y_{({\varepsilon}_{2},{\varepsilon}_{1})}\cup\Phi^{*}_{({\varepsilon}_{1},t)},$

where the superposed point processes are independent. This yields that
$Y_{({\varepsilon}_{2},{\varepsilon}_{1})}\stackrel{{\scriptstyle
D}}{{=}}\Phi^{*}_{({\varepsilon}_{2},{\varepsilon}_{1})}$. Intuitively, this
means that the sequence
$((\hat{y}_{{\varepsilon}_{1}}^{(i)},t^{(i)}),\,i\in\mathbb{Z})$ pertaining to
the process of zero cells until time ${\varepsilon}_{1}$ is consistent with
the Poisson point process properties, which are characteristic for the
consecutive division of cell volumes.

Because this holds for all ${\varepsilon}_{2}>{\varepsilon}_{1}>0$, it is
shown that $Y_{t}=((y_{t}^{(j)},t^{(j)}),\,j\in\mathbb{Z})$ is a Poisson point
process on $\mathbb{R}\times(0,t)$ with intensity measure
$\lambda\otimes\lambda_{(0,t)}$, where $\lambda_{(0,t)}$ denotes the
restriction of the Lebesgue measure to the interval $(0,t)$. Hence the point
process $(y_{t}^{(j)},\,j\in\mathbb{Z})$ of points projected onto
$\mathbb{R}$, is a homogeneous Poisson point process with intensity measure
$t\,\lambda$, and therefore the length of the typical interval between two
points is exponentially distributed with parameter $t$, while the length of
the interval containing the origin $o$ has a gamma-distribution with parameter
$(2,t)$. $\Box$

Cell division processes are potential models for real crack or fracture
pattern appearing in materials science, geology, soft matter or
nanotechnology. The examples considered in [19, 11] illustrate, that a useful
and simple initial goodness-of-fit criterion for such models is the
coefficient of variation CV of the volume of the typical cell. And it became
obvious that this CV of STIT is too large compared to the CV for data of real
crack pattern.

###### Corollary 2

With the assumptions of Lemma 3 for all $t>0$

$\textup{CV}(V_{d}(z)):=\frac{\sqrt{\textup{Var}(V_{d}(z))}}{\mathbb{E}V_{d}(z)}=1,$

where $z$ is a random cuboid with the distribution of the typical cell of
$T_{t}$ and Var denotes the variance of a random variable.

For the cell division process with (L-$\Lambda$) (D-$\Lambda$), which is the
STIT tessellation process, at any time $t>0$ the typical cell has the same
distribution as typical cell of Poisson hyperplane tessellation for the same
$\Lambda$ at the same time, see [20]. Hence, for the translation invariant
measure $\Lambda$ satisfying (1) where $\varphi$ is a Mondrian directional
distribution (13), the CV of the volume of the typical cell equals
$\textup{CV}\left(\prod_{j=1}^{d}Z_{j}\right)$, where $Z_{1},\ldots,Z_{d}$ are
independent exponentially distributed random variables with the parameters
$p_{1}t,\ldots,p_{d}t$, respectively. Thus

$\textup{CV}\left(V_{d}\left(z^{\textup{STIT}}\right)\right)=2^{d}-1,$

where $z^{\textup{STIT}}$ is a random cuboid with the distribution of the
typical cell of the STIT tessellation.

Note that these CV do neither depend on the time $t$ nor on the probabilities
$p_{1},\ldots,p_{d}$. Obviously, for $d\geq 2$ the variability of the volume
of the typical cell of the (L-$V_{d}$) (D-$\Lambda$) cell division process is
considerably smaller than that one of STIT.

###  5.3 A cell division process in a bounded window $W$ and its relation to
fragmentation

Now consider a fixed window $W\in{{\cal P}_{d}}$ which is a cuboid with its
sides parallel or orthogonal to all axes of $\mathbb{R}^{d}$. An (L-$V_{d}$)
(D-$\Lambda$) cell division process $(T_{W,t},\,t\geq 0)$ within $W$ is
launched with $T_{W,0}:=\\{W\\}$. Note that its distribution is different from
$(T_{t}\wedge W,\,t>0)$, which is the restriction to $W$ of the (L-$V_{d}$)
(D-$\Lambda$) cell division process$(T_{t},\,t>0)$ in the whole
$\mathbb{R}^{d}$.

The process $(T_{W,t},\,t\geq 0)$ can be related to a particular case of a
conservative fragmentation process as described in [1].

Assume that $G(W)=1$. For $t>0$ and $T_{W,t}=\\{z_{1},\ldots,z_{n}\\}$ define

${\bf G}(T_{W,t}):=(G(z_{i_{1}}),\ldots,G(z_{i_{n}}),0,0,\ldots)$

such that $\\{i_{1},\ldots,i_{n}\\}=\\{1,\ldots,n\\}$ and
$G(z_{i_{1}})\geq\ldots\geq G(z_{i_{n}})$.

If we choose $G(z)=V_{d}(z)$ for $z\in{{\cal P}_{d}}$, then the (L-$V_{d}$)
(D-$\Lambda$) cell division process $(T_{W,t},\,t\geq 0)$ induces a pure jump
Markov process $({\bf G}(T_{W,t}),\,t\geq 0)$ on
${\mathfrak{S}}:=(0,\infty)^{\mathbb{N}}$ with the following properties.

  1. (a)

$\sum_{z\in T_{W,t}}G(z)=1$ for all $t\geq 0$.

  2. (b)

The holding time of a state $(G(z_{i_{1}}),\ldots,G(z_{i_{n}}),0,0,\ldots)$ is
exponentially distributed with parameter 1.

  3. (c)

At the $n$-th jump time $\tau_{n}$, $n\in\mathbb{N}$, the process jumps from a
state
$(G(z_{i_{1}}),\ldots,G(z_{i_{n}}),0,0,\ldots)$ to
$(G(z_{j_{1}}),\ldots,G(z_{j_{n+1}}),0,0,\ldots)$ by the following rule:
The index $i_{k}$, $k\in\\{1,\ldots,n\\}$, is chosen at random with
probability
$G(z_{i_{k}})=G(z_{i_{k}})/\sum_{\ell\in\\{1,\ldots,n\\}}G(z_{i_{\ell}})$.
Then the cell $z_{i_{k}}$ is divided into two new fragments,
$z^{\prime}_{i_{k}},z^{\prime\prime}_{i_{k}}$ say, and

$\left(G(z^{\prime}_{i_{k}}),G(z^{\prime\prime}_{i_{k}})\right)=\left(U\,G(z_{i_{k}}),(1-U)\,G(z_{i_{k}})\right),$
(19)

where $U$ is a random variable, uniformly distributed on $(0,1)$ and
independent of all the other random variables considered here.

  4. (d)

Putting the $G$-values of the $n+1$ cells into an descending order yields the
new state $(G(z_{j_{1}}),\ldots,G(z_{j_{n+1}}),0,0,\ldots)$.

This shows that, according to Definition 1.1 of [1], the process $({\bf
G}(T_{W,t}),\,t\geq 0)$ is a self-similar fragmentation chain with index of
self-similarity $\alpha=1$ and the dislocation measure $\nu$ which is defined
by (19), that is, $\nu$ is the law of $(\xi,1-\xi,0,0,\ldots),$ where $\xi$ is
a random variable which is uniformly distributed on the interval
$\left[\frac{1}{2},1\right]$.

Note, if $G$ is chosen as another intrinsic volume $V_{n}$, $n<d$, then
$\sum_{z\in T_{W,t}}G(z)\geq 1$ with ${}^{\prime}>^{\prime}$ after the first
division. Then this is no longer the conservative scenario in the sense of
[1].

##  6 Discussion and open problems

The ’constructive proof’ of the existence result of Theorem 1 of a cell
division process does not imply a uniqueness result. We conjecture that the
distribution of the process is uniquely determined by $G$ and $\Lambda$, but
it is an open problem, whether for (L-$G$) and (D-$\Lambda$) also other
distributions of cell division processes exist. Up to now, only the uniqueness
of the (L-$\Lambda$) (D-$\Lambda$) cell division process is shown, which is
the STIT tessellation process driven by $\Lambda$.

Somehow related to the uniqueness problem is the question whether the
sufficient condition (10) for the existence of an (L-$G$) (D-$\Lambda$) cell
division process is also a necessary one.

In a cell division process $(T_{t},\,t>0)$ the cells $z$ are divided
subsequently as long as $G(z)>0$. Therefore, in general there will not be a
limiting random tessellation for $t\to\infty$. Thus the problem arises whether
there is a scaling of the process which stabilizes the one-dimensional
distributions of the process or even yields a time-stationary process. For a
tessellation $T\in\mathbb{T}$ the scaling by a factor $c>0$ means
$cT:=\\{cz;\,z\in T\\}$, with $cz:=\\{cx:\,x\in z\\}$.

For the STIT tessellation process $(Y_{t},\,t>0)$ driven by an arbitrary
$\Lambda$ satisfying (1) and (2) it was shown in [20] that
$t\,Y_{t}\stackrel{{\scriptstyle D}}{{=}}Y_{1}\ \textup{ for all }t>0.$
Furthermore, in [12] it was proven that for $a>1$ the process
$a^{t}\,Y_{a^{t}}$ is stationary in time.

For (L-$V_{d}$) (D-$\Lambda$) and the Mondrian directional distribution, by
Lemma 3, for all $t>0$ and for the zero cell $z^{0}_{t}$ of $T_{t}$ we have
$t\,V_{d}(z^{0}_{t})\stackrel{{\scriptstyle D}}{{=}}V_{d}(z^{0}_{1})$ and
hence $V_{d}(t^{1/d}z^{0}_{t})\stackrel{{\scriptstyle
D}}{{=}}V_{d}(z^{0}_{1}).$ This does not yet show a scaling property for the
whole random tessellation $T_{t}$ but one can see that, if there is a scaling
factor, it can only be $t^{1/d}$.

Regarding tail triviality and other ergodic properties of the tessellations
generated by cell division, the method of the so-called encapsulation used in
our earlier papers [13, 14], cannot be applied here directly for the models
which are not spatially consistent. But the method used in the proof of Lemma
1 for zero-cells can be developed further.

Real crack structures, some of them mentioned in the introduction, show the
tendency that a crack of a cell divides it close to the center. This is not
yet incorporated in the models with a (D-$\Lambda$) division rule where
$\Lambda$ is a translation invariant measure. Maybe that this can partially be
compensated in some cases using the life time rule (L-$G^{\alpha}$) instead of
(L-$G$), with a large value of the exponent $\alpha$, see also Remark 1. But
nevertheless, it is desirable to develop models for cell division processes
with alternative division rules as well as life time distributions. Here, a
combination of the results of [28, 4] with the present approach could be
promising. Simulation studies for tessellations in a bounded window of the
plane $\mathbb{R}^{2}$ with several life time rules L and also a variety of
division rules D, aimed to adapt models to real structures, are presented in
[11, 10]. Codes for a simulation of those tessellations are available in [9].

## Acknowledgment

This work was supported by the Center for Mathematical Modeling ANID Basal
FB210005. In particular, Werner Nagel is indebted for the support of his
visits at the Center, where in March 2023 this paper could be finished.

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