11institutetext:  Katarzyna Pichór 22institutetext: Institute of Mathematics,
University of Silesia in Katowice, Bankowa 14, 40-007 Katowice, Poland,
22email<EMAIL_ADDRESS>33institutetext: Ryszard Rudnicki
44institutetext: Institute of Mathematics, Polish Academy of Sciences, Bankowa
14, 40-007 Katowice, Poland, 44email<EMAIL_ADDRESS>
# Transformation semigroups and their applications

Katarzyna Pichór and Ryszard Rudnicki

###### Abstract

In this chapter we present transformation semigroups and their applications.
We begin with Klein’s approach to geometry based on invariants of
transformation groups. Then we present symmetry groups in chemistry and in
classical mechanics. Next we introduce one-parameter semigroups of
transformations and their applications in ergodic theory. Our main subject are
one-parameter semigroups of operators, in particular stochastic semigroups. We
present general results on their existence and long-time behaviour. We also
give examples of one-parameter semigroups related to Markov chains, diffusion
and processes with jumps. We focus on the applications of semigroups of
operators in biology. Among other things, we study models of: DNA evolution;
growth of erythrocyte population; gene expression; cell cycle; the movement of
bacteria and insects. We also consider models with stochastic noise and
different population models.

###### Keywords:

transformation semigroup, stochastic semigroup, Markov process, asymptotic
stability, biological model

##  1 Introduction

Transformation semigroups are used today in many scientific fields because
they allow a unified description of various natural processes and convenient
formulation of theorems concerning their properties. The intensive study of
semigroups began in the early XX century as a generalization of the notion of
a group. The concept of a group was introduced by Évariste Galois in the 1830s
to study polynomial equations. The abstract definition of a group was
formulated by Arthur Cayley Cayley  in 1854 but, as in the case of Galois, it
referred only to finite groups. In 1872 Felix Klein in his famous the Erlangen
program presented a novel approach to geometry based on the invariants of
transformation groups (Sec. 2.2).

Transformation groups are used to study symmetries of some geometrical or
chemical objects. In Sec. 2.3 we present a short introduction to molecular and
crystal symmetries. The idea of using transformation groups has been
successfully applied to describe Newtonian mechanics Arnold . In Sec. 2.4 we
present the group of Galilean transformations which, in combination with
Newton’s principle of determinacy, allow the derivation of equations of motion
in inertial systems; and the fundamental laws of classical mechanics.

One-parameter semigroups of transformations of some space $X$ are called
dynamical systems. Different types of dynamical systems are studied depending
on the structure of the space $X$. For example if $X$ is a topological space,
then we study continuous dynamical systems on $X$; if $X$ is a differentiable
manifold — smooth dynamical systems. In ergodic theory measure preserving
dynamical systems are considered. Such systems consist of measurable
transformations on a probability space $(X,\Sigma,\mu)$ invariant with respect
to the measure $\mu$ CFS ; Krengel ; LiM ; PM . Properties of dynamical
systems can be studied using their representations on the space
$L^{1}(X,\Sigma,\mu)$. In Sec. 2.5 we introduce a semigroup of
Frobenius–Perron operators on $L^{1}$ and we show the relations between
ergodic properties of a dynamical system and asymptotic behavior of the
semigroup of Frobenius–Perron operators.

Among the transformation semigroups, the one-parameter semigroups of operators
play an important role, as they are used to describe and study processes that
change over time. They are usually associated with evolution equations of the
form $x^{\prime}(t)=Ax(t)$, where $A$ is some linear (often unbounded)
operator in a Banach space. We introduce in Sec. 3.1 the notion of
$C_{0}$-semigroups of operators and we show their relation with evolution
equations.

Our main subject are stochastic semigroups. Stochastic semigroups are
semigroups of positive linear operators defined on the space $L^{1}$
preserving the set of densities. These semigroups are generated by partial
differential equations of different types and describe the behaviour of the
distributions of Markov processes like diffusion processes, piecewise
deterministic Markov processes (PDMPs) and hybrid stochastic processes
Rudnicki-LN . They are also used to study ergodic properties of dynamical
systems LiM . In Sec. 3.2 we introduce the notions of stochastic and
substochastic semigroups and we state general theorems on their existence. In
the following sections, we present basic examples of stochastic semigroups
related to: Markov chains (Sec. 3.3), diffusion (Sec. 3.4), processes with
jumps (Sec. 3.5), processes with random switching (Sec. 3.6), and to
stochastic billiards (Sec. 3.7).

Long-time behaviour of semigroups of operators is studied in Part 4. We begin
with a continuous version of the classical Perron–Frobenius theorem (Sec.
4.1). This theorem concerns semigroups of linear operators on the space
$\mathbb{R}^{n}$. The study of such semigroups is a good starting point for
introducing asymptotic properties of semigroups in Banach spaces and provides
simple examples. One of the tools in studying the behaviour of a quasi-compact
semigroup of operators when $t\to\infty$ is its spectral decomposition (see
Sec. 4.2). In particular, if the semigroup $\\{T(t)\\}_{t\geq 0}$ on a Banach
lattice $E$ is also positive and irreducible, then there exist
$r\in\mathbb{R}$, nonzero $x^{*}\in E$, and $\alpha\in E^{*}$ such that

$\lim_{t\to\infty}e^{-r\,t}T(t)x=x^{*}\alpha(x)\quad\textrm{for $x\in E$}.$
(1)

Asymptotic properties of stochastic semigroups are studied in the next four
sections. We begin with a theorem concerning uniformly continuous stochastic
semigroups and with the lower function theorem of Lasota and Yorke LY  (see
Sec. 4.3). The main results concern stochastic semigroups bounded from below
by integral operators called partially integral semigroups (see Sec. 4.4). If
such a semigroup $\\{P(t)\\}_{t\geq 0}$ has a unique and positive density
$f^{*}$, then it is asymptotically stable, i.e. $P(t)f\to f^{*}$ for each
density $f$. Strengthening slightly the condition for partial integrability,
we obtain a theorem about the decomposition of a semigroup into asymptotically
stable parts and a sweeping part (see Sec. 4.5). This theorem and its
conclusions are convenient for the study of many advanced stochastic
semigroups. We show in Sec. 4.6 that if a stochastic semigroup has some
compact properties then it has also an asymptotic decomposition.

Part 5 is devoted to applications in biological models. Our aim is to
demonstrate both the wealth of applications of semigroups in biological models
as well as the usefulness of the methods presented for their study. Among
other things, we study models of: DNA evolution; growth of erythrocyte
population; population growth with stochastic noise; gene expression; the
movement of bacteria and insects. We also study structured population models
and more advanced models based on semi-Markov processes – kangaroo movement
and a cell cycle model PR-cell-2022 .

##  2 Transformation semigroups everywhere

###  2.1 Basic definitions and examples

A semigroup $(\mathbf{G},\cdot)$ is an arbitrary set $\mathbf{G}$ together
with a binary operation $\cdot$ on $\mathbf{G}$ with the property of
associativity, i.e. $(x\cdot y)\cdot z=x\cdot(y\cdot z)$ for all
$x,y,z\in\mathbf{G}$. An element $e\in\mathbf{G}$ is called identity if
$e\cdot x=x\cdot e=x$ for all $x\in\mathbf{G}$. From the property of
associativity it follows that a semigroup has at most one identity element. A
monoid is a semigroup with an identity. An element $x$ of the monoid
$(\mathbf{G},\cdot,e)$ has two-sided inverse $x^{-1}\in\mathbf{G}$, shortly
called inverse, if $x\cdot x^{-1}=x^{-1}\cdot x=e$. A monoid is called a group
if each element of $\mathbf{G}$ has an inverse. A semigroup
$(\mathbf{G},\cdot)$ is called commutative if $x\cdot y=y\cdot x$ for all
$x,y\in\mathbf{G}$.

A semigroup of elements acting as transformations of some space $X$ is called
a transformation semigroup or a semigroup action on a set. In this case the
binary operation on $\mathbf{G}$ is the composition $\circ$ of transformations
and the identity element is the identity transformation $I$.

A topological semigroup $(\mathbf{G},\cdot)$ is a topological space that is
also a semigroup such that the semigroup operation $(x,y)\mapsto x\cdot y$ is
continuous. If $(\mathbf{G},\cdot)$ is a group, then it is called a
topological group if we additionally assume that the inversion map $x\mapsto
x^{-1}$ is continuous.

One-parameter semigroup of transformations is a family
$\mathbf{G}=\\{g_{t}\\}_{t\in T}$ of transformations of a set $X$ such that
$T$ is one of the sets $\mathbb{N}=\\{0,1,\dots\\}$, $\mathbb{Z}$,
$\mathbb{R}$, $\mathbb{R}_{+}=[0,\infty)$; $g_{s}\circ g_{t}=g_{s+t}$ for
$s,t\in T$ and $g_{0}=I$. If $X$ is a topological space and the map
$\varphi\colon T\times X\to X$ given by $\varphi(t,x)=g_{t}(x)$ is continuous,
then family $\mathbf{G}$ is called a $C_{0}$-semigroup. If $T=\mathbb{N}$ then
there is a function $g$ such that $g_{t}$ is the $t$-th iterate of $g$ and if
$T=\mathbb{Z}$ then additionally $g_{-t}=g_{t}^{-1}$ for $\,t>0$. If
$T=\mathbb{N}$ and $g\colon X\to X$ is a continuous function, then
$\mathbf{G}$ is a $C_{0}$-semigroup; if $T=\mathbb{Z}$ and $g$ is a
homeomorphism of $X$, then $\mathbf{G}$ is a $C_{0}$-group. One-parameter
semigroup of transformations is also called a dynamical system. In the theory
of dynamical systems a one-parameter semigroup is called: endomorphism if
$T=\mathbb{N}$; automorphism if $T=\mathbb{Z}$; flow if $T=\mathbb{R}$; and
semiflow if $T=\mathbb{R}_{+}$. If $X$ has some additional structure then a
dynamical system should be compatible with this structure, e.g. if $X$ is
equipped with a topology, then $\mathbf{G}$ should be a $C_{0}$-semigroup.

Transformation semigroups provide a rich source of many examples and counter-
examples of the concepts introduced earlier. The basic examples are linear
maps of $\mathbb{R}^{n}$ or $\mathbb{C}^{n}$. Such a map is of the form
$x\mapsto Ax$ for some $A\in{\rm M}^{n}(K)$, where ${\rm M}^{n}(K)$ denotes
the set of all $n\times n$ matrices over the field $K=\mathbb{R}$ or
$K=\mathbb{C}$, respectively. Since the composition of linear maps corresponds
to the matrix multiplication, the set ${\rm M}^{n}(K)$ with operation of
multiplication and with identity matrix is a monoid. Since ${\rm M}^{n}(K)$ is
isomorphic to $K^{n^{2}}$ and multiplication is a continuous function, ${\rm
M}^{n}(K)$ is a topological semigroup. The subset of ${\rm M}^{n}(K)$ that
consists of all invertible matrices denoted by ${\rm GL}(n,K)$ is a
topological group because the operation $A\mapsto A^{-1}$, $A\in{\rm
GL}(n,K)$, is continuous. By ${\rm O}(n)$ we denote the subset of ${\rm
GL}(n,\mathbb{R})$ that consists of all orthogonal matrices, i.e. $A$ is
orthogonal if $A^{\mathrm{T}}=A^{-1}$, where $A^{\mathrm{T}}$ is the transpose
of $A$. The set ${\rm O}(n)$ is also a topological group. Transformations
corresponding to orthogonal matrices are linear isometries of
$\mathbb{R}^{n}$.

The simplest example of a $C_{0}$-semigroup is a translation semigroup on
$\mathbb{R}$ or $\mathbb{R}_{+}$. The transformations are given by the formula
$g_{t}(x)=x+t$, where $t\in\mathbb{R}$ if $X=\mathbb{R}$ and $t\geq 0$ if
$X=\mathbb{R}_{+}$. Observe that if $X=\mathbb{R}$, then the translation
semigroup $\\{g_{t}\\}$ is a flow and if $X=\mathbb{R}_{+}$ it is a semiflow.
Another useful example of a $C_{0}$-semigroup is the rotation group on a unit
circle $S^{1}$ on a complex space $\mathbb{C}$. This group consists of all
transformations of the form $g_{t}(z)=ze^{it}$, $t\in\mathbb{R}$.

More general example of a $C_{0}$-semigroup can be defined by using systems of
ordinary differential equations. Consider a moving particle in an open set
$X\subset\mathbb{R}^{d}$. We assume that if a particle is at point $x$ then
its velocity is $b(x)$, where $b\colon X\to\mathbb{R}^{d}$ is a differentiable
function. It means that if $x(t)$ is its position at time $t$ then the
function $x(t)$ satisfies the following equation

$x^{\prime}(t)=b(x(t)).$ (2)

We also assume that the particle does not leave the set $X$. For each
$\bar{x}\in X$ we denote by $\pi_{t}\bar{x}$ the solution $x(t)$ of (2) with
the initial condition $x(0)=\bar{x}$. Then $\\{\pi_{t}\\}_{t\geq 0}$ is a
semiflow on $X$. If additionally the solution $x(t)$ exists and $x(t)\in X$
for all $t\in\mathbb{R}$, then $\\{\pi_{t}\\}_{t\in\mathbb{R}}$ is a flow on
$X$.

In Sec. 2.5 we introduce measure-preserving dynamical systems which are one-
parameter semigroups but may not be $C_{0}$-semigroups. Such systems play an
essential role in ergodic theory.

An important role in the study of properties of algebraic structures plays the
notion of homomorphism. If $(\mathbf{G}_{1},\cdot)$ and
$(\mathbf{G}_{2},\circ)$ are semigroups then a homomorphism from
$\mathbf{G}_{1}$ to $\mathbf{G}_{2}$ is a map
$h\colon\mathbf{G}_{1}\to\mathbf{G}_{2}$ such that

$h(g_{1}\cdot g_{2})=h(g_{1})\circ h(g_{2})\quad\text{for all
$\,g_{1},g_{2}\in\mathbf{G}_{1}$}.$ (3)

If $\mathbf{G}_{1}$ and $\mathbf{G}_{2}$ are monoids, then we additionally
assume that a homomorphism maps the identity element of $\mathbf{G}_{1}$ to
the identity element of $\mathbf{G}_{2}$. If $\mathbf{G}_{1}$ and
$\mathbf{G}_{2}$ are groups and $h$ satisfies (3), then the homomorphism $h$
maps the identity to the identity and $h(g^{-1})=(h(g))^{-1}$ for all
$g\in\mathbf{G}_{1}$.

We now give an example of a homomorphism from a semigroup without linear
structure to a semigroup defined on a linear space. We will use the presented
construction in Sec. 2.5. Let $X$ be any set and let $(\mathbf{G},\circ)$ be
the semigroup which consists of all functions from $X$ to $X$ with composition
as the binary operation: $(\varphi,\psi)\mapsto\varphi\circ\psi$. Let
$V=F(X,\mathbb{R})$ be the set of all maps from $X$ to $\mathbb{R}$ and let
$\mathbf{L}=L(V,\mathbb{R})$ be the set of all linear operators on the space
$V$. The space $\mathbf{L}$ is a linear space but it is also a semigroup with
the binary operation: $(P,Q)\mapsto Q\circ P$. Given a function $\varphi\colon
X\to X$ we define a linear operator $U_{\varphi}\in\mathbf{L}$ by the formula
$(U_{\varphi}f)(x)=f(\varphi(x))$ for $f\in V$. Then

$U_{\varphi\circ\psi}=U_{\psi}\circ U_{\varphi},$

which means that the map $h\colon\mathbf{G}\to\mathbf{L}$ given by
$h(\varphi)=U_{\varphi}$ is a homomorphism. The homomorphism $h$ is a
representation of the group $\mathbf{G}$. Generally, if $(\mathbf{G},\cdot)$
is an arbitrary semigroup, $V$ is a linear space and ${\rm End}(V)$ is the set
of all linear transformations (endomorphisms) $V\to V$, and
$h\colon\mathbf{G}\to{\rm End}(V)$ is a homomorphism then $h$ is called a
representation of the semigroup $\mathbf{G}$ on the vector space $V$.

The operator $U_{\varphi}$ is called the Koopman operator or the composition
operator related to $\varphi$. Usually the set $X$ has an additional
structure. If for example $(X,\Sigma)$ is a measurable space, then we can
consider a subgroup of $\mathbf{G}$ consisting of all measurable
transformations. Then the space $V$ can be the set of all measurable functions
from $X$ to $\mathbb{R}$.

If instead of the above semigroup $\mathbf{G}$ we consider the group $({\rm
S}(X),\circ)$ of all invertible functions from $X$ onto $X$, then the
homomorphism $h$ transfers ${\rm S}(X)$ into the set of invertible linear
transformations ${\rm GL}(V)$, $V=F(X,\mathbb{R})$, and the homomorphism $h$
is a representation of the group ${\rm S}(X)$. In general, if
$(\mathbf{G},\cdot)$ is an arbitrary group, $V$ is a linear space, ${\rm
GL}(V)$ is the set of bijective linear transformations from $V$ to $V$, and
$h\colon\mathbf{G}\to{\rm GL}(V)$ is a homomorphism, then $h$ is called a
representation of the group $\mathbf{G}$ on the vector space $V$.

###  2.2 Transformation groups in geometry

As we have mentioned in Introduction, different geometries can be classified
by transformation groups. Consider a given group $\mathbf{G}$ of
transformations of the space $\mathbb{R}^{3}$. Klein proposed that a geometry
related to the group $\mathbf{G}$ consists of the objects (properties or
figures) which, as a result of the action of any transformation
$S\in\mathbf{G}$, transfer to objects of the same type. Euclidean geometry
corresponds to the isometric transformations (translations, rotations,
reflections and their compositions). We recall that any isometric
transformation is given by the formula $Sx=Ax+b$, where $b\in\mathbb{R}^{3}$
and the matrix $A$ is orthogonal. The objects of Euclidean geometry are for
example: the distance, perpendicularity and parallelity of straight lines,
circles, preservation of angles.

If we replace the group of isometries by the group of similarities, i.e. the
transformations of the form $Sx=cAx+b$, where $c\neq 0$ is a real constant and
$A$ is an orthogonal matrix, then we obtain another geometry that Euclid
actually dealt with. This geometry has the same aforementioned objects as
Euclidean geometry with the exception of distance.

Affine geometry is the geometry of affine transformations, i.e. the maps of
the form $Sx=Ax+b$, where $A$ is any invertible linear transformation. The
objects in affine geometry are parallelity of straight lines and ratios of
lengths.

More advanced is the projective geometry which is induced by homographies. A
point in the projective space $X$ of dimension $n$ is a one-dimensional
subspace of the vector space $\mathbb{R}^{n+1}$. If
$\mathbf{x}=[x_{0},x_{1},\dots,x_{n}]$ is a non-zero element of
$\mathbb{R}^{n+1}$ then

$[x_{0}:x_{1}:\dots:x_{n}]=\\{[\theta x_{0},\theta x_{1},\dots,\theta
x_{n}]\colon\theta\in\mathbb{R}\\}$

is a one-dimensional subspace of $\mathbb{R}^{n+1}$ spanned by the vector
$\mathbf{x}$. Let

$\mathbb{R}^{n+1}_{0}=\mathbb{R}^{n+1}\setminus\\{\mathbf{0}\\},\quad\mathbb{R}^{n+1}_{0,\bullet}=\mathbb{R}^{n+1}\setminus\\{\mathbf{x}\in\mathbb{R}^{n+1}\colon
x_{0}=0\\}.$

Then we can consider a map $P\colon\mathbb{R}^{n+1}_{0}\to X$ given by

$P([x_{0},x_{1},\dots,x_{n}])=[x_{0}:x_{1}:\dots:x_{n}].$

Let $A\colon\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$ be an invertible linear map.
A homography, called also projective transformation, induced by $A$ is a
mapping $f_{A}$ from $X$ to $X$ given by

$f_{A}([x_{0}:x_{1}:\dots:x_{n}])=[(A\mathbf{x})_{0}:(A\mathbf{x})_{1}:\dots:(A\mathbf{x})_{n}].$

If $[a_{ij}]$ is a matrix of the map $A$, then it is called the matrix of the
homography $f_{A}$. The following diagram commutes:

$\begin{CD}\mathbb{R}^{n+1}_{0}@>{A}>{}>\mathbb{R}^{n+1}_{0}\\\
@V{P}V{}V@V{}V{P}V\\\ X@>{}>{f_{A}}>X,\end{CD}$

i.e. $f_{A}\circ P=P\circ A$. If $V$ is a linear subspace of dimension
$m+1\leq n$ in $\mathbb{R}^{n+1}$, then $P(V\setminus\\{\mathbf{0}\\})$ is
called a flat of dimension $m$ in $X$. In particular the lines in $X$ are
images under $P$ of two-dimensional linear subspaces. From the formula
$f_{A}\circ P=P\circ A$ it follows that a homography transfers a flat on a
flat of the same dimension.

Figure 1: The map $Q$ transfers the line $l$ to the point $B$.

We can also introduce a projective space in a way that is closer to
geometrical intuition. Let $\pi$ be the hyperplane
$\\{\mathbf{x}\in\mathbb{R}^{n+1}\colon x_{0}=1\\}$. Fix a point
$\mathbf{x}\in\mathbb{R}^{n+1}_{0,\bullet}$. Then the vector
$\overrightarrow{\,\mathbf{x}\mathbf{0}\,}$ spans the line
$l=[x_{0}:x_{1}:\dots:x_{n}]$. Denote by $B$ the intersection of $l$ and $\pi$
(see Fig. 1). In this way we obtain a bijective transformation of the set of
all lines which are not parallel to the hyperplane $\pi$ onto the space
$\mathbb{R}^{n}$ (we omit $1$ \- the first coordinate of the point $B$). The
projective space $Y$ is defined by adding the set $Y_{\infty}$ of ”points at
infinity” to $\mathbb{R}^{n}$, which corresponds to lines parallel to the
hyperplane $\pi$. Thus $Y=\mathbb{R}^{n}\cup Y_{\infty}$. In one-dimensional
case $Y_{\infty}$ is a single point, so we can write
$Y=\mathbb{R}\cup\\{\infty\\}$. If $n=2$, then $\mathbb{R}^{2}$ and
$Y_{\infty}$ can be identified with a complex plane $\mathbb{C}$ and with a
semicircle at infinity, respectively, so we can write
$Y_{\infty}=\\{\infty\cdot e^{it}\colon t\in[0,\pi)\\}$. For arbitrary $n$ the
set $Y_{\infty}$ is an $(n-1)$-dimensional hemisphere at infinity, precisely,
the hemisphere is formed by identification of antipodal points on the sphere.

Let $\mathbf{x}\in\mathbb{R}^{n+1}_{0,\bullet}$. Then
$B=\big{(}1,\frac{x_{1}}{x_{0}},\dots,\frac{x_{n}}{x_{0}}\big{)}$ and we can
define the map $Q\colon\mathbb{R}^{n+1}_{0,\bullet}\to\mathbb{R}^{n}$ by the
formula
$Q(x_{0},x_{1},\dots,x_{n})=\big{(}\frac{x_{1}}{x_{0}},\dots,\frac{x_{n}}{x_{0}}\big{)}$.
It is not difficult to check that if $V$ is an $(m+1)$-dimensional subspace of
$\mathbb{R}^{n+1}$ such that
$V\setminus\\{\mathbf{0}\\}\subset\mathbb{R}^{n+1}_{0,\bullet}$, then
$Q(V\setminus\\{\mathbf{0}\\})$ is an $m$-dimensional flat in
$\mathbb{R}^{n}$, i.e. $Q(V\setminus\\{\mathbf{0}\\})=W+\mathbf{b}$, where $W$
is an $m$-dimensional subspace of $\mathbb{R}^{n}$ and
$\mathbf{b}\in\mathbb{R}^{n}$.

In this model the homography $f_{A}$ restricted to the set $\mathbb{R}^{n}$ is
given by the formula

$f_{A}(x_{1},\dots,x_{n})=(y_{1},\dots,y_{n}),\quad\text{where
$y_{k}=\frac{a_{k0}+a_{k1}x_{1}+\dots+a_{kn}x_{n}}{a_{00}+a_{01}x_{1}+\dots+a_{0n}x_{n}}$,\quad$k=1,\dots,n$,}$

for all points $(x_{1},\dots,x_{n})$ such that the last denominator differs
from $0$, otherwise the point is transfered on a point at infinity. The group
of all homographies contains affine maps and central perspectivities. It is
interesting that though a homography is generally a nonlinear map, it
transfers a flat on a flat, but does not preserve parallelity. Since the group
of homographies is larger than the group of affine maps all theorems of
projective geometry remain valid in affine geometry.

It should be noted that $n$-dimensional projective space can be treated as a
compact smooth manifold $M$ and construct as the closed $n$-dimensional ball
with identified antipodal points on the boundary. The map
$h(\mathbf{x})=\frac{\mathbf{x}}{1+\|\mathbf{x}\|}$ transfers part
$\mathbb{R}^{n}$ of $Y$ onto the open ball $B(\mathbf{0},1)$ and the infinite
points are transfered on the boundary of $M$. When $n=1$ then the set
$Y_{\infty}$ is a one-point set, thus the projective space is topologically
equivalent to a circle. In $n=2$ the manifold $M$ cannot be embedded in
$\mathbb{R}^{3}$.

The idea of using transformation groups to define invariants has found
applications in other branches of geometry, for example in elliptic geometry.
In topology that has separated from geometry, we consider objects invariant
with respect to a group of homeomorphisms. Moreover, homotopy groups and
cohomotopy groups play important role in algebraic topology.

###  2.3 Symmetry groups in chemistry

Transformation groups are also applied to study symmetries of some geometrical
or chemical objects. Planar or spatial figures are invariant with respect to
some transformations called symmetries. All symmetries of a given figure $X$
form a subgroup of ${\rm O}(n)$, $n=2,3$, called a symmetry group and denoted
by ${\rm Sym}(X)$. For example, if $X$ is a non-square rectangle then this
group consists of four symmetries: identity $I$, two reflections about the
horizontal and vertical middle lines $S_{h}$, $S_{v}$ and the central symmetry
$S_{O}$. The map $h\colon{\rm Sym}(X)\to\\{0,1,2,3\\}$ given by the formulae:
$h(I)=0$, $h(S_{h})=1$, $h(S_{v})=3$, $h(S_{O})=2$, is the homomorphism
between the groups ${\rm Sym}(X)$ and $(\mathbb{Z}_{4},\oplus)$, where
$\mathbb{Z}_{4}=\\{0,1,2,3\\}$ and $a\oplus b=(a+b)\mod 4$.

In chemistry symmetry groups describe molecular and crystal symmetries. Group
theory enables simplification of analysis of chemical and physical properties
of molecules and crystals. Molecular symmetries are operations performed on
chemical molecules leading to an overlapping arrangement of the molecule’s
atoms before and after the operation.

Figure 2: Rotation $S_{l}$ around the axis $l$ by an angle of $120^{\circ}$
does not change an ammonia molecule. The mirror plane $\sigma_{2}$ is marked
on the right picture.

As an example consider an ammonia molecule ${\rm{NH}}_{3}$ (see Fig. 2). Let
$l$ be a straight line passing through the nitrogen atom and the centre of the
triangle defined by the hydrogen atoms. Then the symmetry group of this
molecule consists of six isometries: identity $I$; two rotations
$S_{1}=S_{l}$, $S_{2}=S_{l}^{2}$ around the axis $l$; and three reflections
$S_{\sigma_{1}}$, $S_{\sigma_{2}}$, $S_{\sigma_{3}}$. Here $S_{l}$ is the
rotation around the axis $l$ by an angle of $120^{\circ}$. The mirror planes
$\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$ ran through the axis $l$ and one of
the hydrogen atoms. The symmetry group of ${\rm NH}_{3}$ is denoted by ${\rm
C}_{\rm 3v}$ (see Tab. 1). It is obvious that this group is isomorphic to the
symmetry group of the equilateral triangle.

$g$ $f$ | $I$ | $S_{l}$ | $S_{l}^{2}$ |  $S_{\sigma_{1}}$ |  $S_{\sigma_{2}}$ |  $S_{\sigma_{3}}$
---|---|---|---|---|---|---
$I$ | $I$ | $S_{l}$ | $S_{l}^{2}$ | $S_{\sigma_{1}}$ | $S_{\sigma_{2}}$ | $S_{\sigma_{3}}$
$S_{l}$ | $S_{l}$ | $S_{l}^{2}$ | $I$ | $S_{\sigma_{3}}$ | $S_{\sigma_{1}}$ | $S_{\sigma_{2}}$
$S_{l}^{2}$ | $S_{l}^{2}$ | $I$ | $S_{l}$ | $S_{\sigma_{2}}$ | $S_{\sigma_{3}}$ | $S_{\sigma_{1}}$
$S_{\sigma_{1}}$ | $\,S_{\sigma_{1}}$ |  $S_{\sigma_{2}}$ | $\,S_{\sigma_{3}}$ | $I$ | $S_{l}$ | $S_{l}^{2}$
$S_{\sigma_{2}}$ | $S_{\sigma_{2}}$ | $S_{\sigma_{3}}$ | $S_{\sigma_{1}}$ | $S_{l}^{2}$ | $I$ | $S_{l}$
$S_{\sigma_{3}}$ | $S_{\sigma_{3}}$ | $S_{\sigma_{1}}$ | $S_{\sigma_{2}}$ | $S_{l}$ | $S_{l}^{2}$ | $I$
Table 1: The results of the operation $g\circ f$ in the group ${\rm C}_{\rm
3v}$.

A group of transformations of the spaces $\mathbb{R}^{n}$ is called a point
group if all transformations have a common fixed point. All isometries of a
bounded object are point groups. Thus, a molecular symmetry group is also a
point group. A crystallographic group is the group of all isometries that
leave the crystal structure unchanged, i.e. the same kinds of atoms would be
placed in similar positions as before the transformation. A crystallographic
group includes some translations of the space. A crystallographic point group
is a subgroup of the crystallographic group consisting of all transformations
having a common fixed point. Thought there are infinitely many three-
dimensional point groups, there are only 32 crystallographic point groups.

The crystallographic point group determines, the directional variation of
physical properties that arise from its structure, including optical and
electro-optical features. The invariance of a crystal with respect to a
translational defines crystallinity. Other applications of symmetry groups in
quantum mechanical and in the spectroscopy can be found in Blinder ; Carter ;
Cotton .

###  2.4 Applications to Newtonian mechanics

We will study the motion of material points in a three-dimensional Euclidean
space. The position of points changes with time, so it is natural to assume
that time is an element of space. Thus, physical space, called world, is a
four-dimensional space. Since in this space we do not distinguish the origin
of the coordinate system, the world is considered to be a four-dimensional
affine space $A^{4}=A^{1}\times A^{3}$, in which $A^{1}$ is identified with
the time axis, and at each fixed time $t$ a set of points (events) of the form
$\\{t\\}\times A^{3}$ is called subspace of simultaneous events. The
difference of points from the space $A^{3}$ is a vector from $\mathbb{R}^{3}$,
so in the subspace of simultaneous events it is possible to introduce a
distance

$\varrho((t,\mathbf{a}),(t,\mathbf{b}))=\|\mathbf{a}-\mathbf{b}\|=\sqrt{\smash[b]{(\mathbf{a}-\mathbf{b},\mathbf{a}-\mathbf{b})}},$

where $\|\cdot\|$ , $(\cdot,\cdot)$ is respectively the norm and the scalar
product in $\mathbb{R}^{3}$.

Many coordinate systems can be introduced in the space $A^{4}$, among which we
distinguish inertial systems. These systems are characterised by Galileo’s
principle of relativity:

  1. (1)

all the laws of nature are the same at all moments of times in all inertial
coordinate systems,

  2. (2)

each coordinate system moving in uniform rectilinear motion with respect to an
inertial system is also an inertial system.

We also assume Newton’s principle of determinacy, which states that the
positions and velocities of the points of the system at the initial moment
uniquely determine the entire motion of the system.

We introduce the coordinate system in the space
$\mathbb{R}\times\mathbb{R}^{3}$. Then the motion of a single point can be
described by the function $\mathbf{x}\colon I\to\mathbb{R}^{3}$, where $I$ is
the interval on the time axis. The set $\mathbf{x}(I)$ is called the
trajectory of a point, and the graph of the mapping, i.e. the set
$\\{(t,\mathbf{x}(t))\colon t\in I\\}$, is called the world line.

The following transformations of the space $\mathbb{R}\times\mathbb{R}^{3}$
have the property that the world lines of any mechanical system subjected to
the same transformation pass to other world lines of the same system:

  1. (a)

uniform motion with velocity $\mathbf{v}\in\mathbb{R}^{3}$

$g_{1}(t,\mathbf{x})=(t,\mathbf{x}+\mathbf{v}t),$

  2. (b)

translation of the origin $(s,\mathbf{a})\in\mathbb{R}\times\mathbb{R}^{3}$

$g_{2}(t,\mathbf{x})=(t+s,\mathbf{x}+\mathbf{a}),$

  3. (c)

orthogonal transformation (rotation or plane symmetry)

$g_{3}(t,\mathbf{x})=(t,G\mathbf{x}),$

where $G:\mathbb{R}^{3}\to\mathbb{R}^{3}$ is an orthogonal transformation. The
set of all possible compositions of transformations of types
$g_{1},g_{2},g_{3}$ forms the group of Galilean transformations. It can be
easily checked that any Galilean transformation is of the form $g_{1}\circ
g_{2}\circ g_{3}$. Preserving the world line by the $g_{1}$ transformation
follows from Galileo’s principle of relativity, while by the transformations
$g_{2}$ and $g_{3}$ from the geometrical structure of space: invariance to
displacements (homogeneity) and rotations (isotropy).

It is possible to derive the general form of the equations of motion of
material points from Newton’s principle of determinacy. Since the motion of
points is uniquely determined by their initial positions and velocities, so
the acceleration at time $t$ is a function of the position and velocity also
at time $t$. Consider $n$ material points. Let $\mathbf{x}_{i}(t)$ describes
the position of the $i$-th point at time $t$. Then the function

$\mathbf{x}(t)=(\mathbf{x}_{1}(t),\ldots,\mathbf{x}_{n}(t))$

describes the motion of a system of $n$ points. Let $N=3n$. Since acceleration
is a function of position and velocity, so there exists a function
$\mathbf{F}:\mathbb{R}^{N}\times\mathbb{R}^{N}\times\mathbb{R}\to\mathbb{R}^{N}$
such that

$\mathbf{x}^{\prime\prime}(t)=\mathbf{F}(\mathbf{x}(t),\mathbf{x}^{\prime}(t),t),$
(4)

where
$\mathbf{x}^{\prime}(t)=(\mathbf{x}_{1}^{\prime}(t),\ldots,\mathbf{x}_{n}^{\prime}(t))$.

We will now show how the invariance of the world line with respect to the
Galilean group imposes certain conditions on the function $\mathbf{F}$. One of
the Galilean transformations is a time shift. This means that if
$\mathbf{x}(t)$ is a solution of Eq. (4), then also for any $s$ the function
$\mathbf{y}(t)=\mathbf{x}(t+s)$ is a solution of Eq. (4). From Eq. (4) we
obtain

$\mathbf{x}^{\prime\prime}(t+s)=\mathbf{F}(\mathbf{x}(t+s),\mathbf{x}^{\prime}(t+s),t+s).$
(5)

The function $\mathbf{y}$ also satisfies Eq. (4), thus

$\mathbf{y}^{\prime\prime}(t)=\mathbf{F}(\mathbf{y}(t),\mathbf{y}^{\prime}(t),t)=\mathbf{F}(\mathbf{x}(t+s),\mathbf{x}^{\prime}(t+s),t).$
(6)

Since $\mathbf{x}^{\prime\prime}(t+s)=\mathbf{y}^{\prime\prime}(t)$, from Eqs.
(5) and (6) it follows that

$\mathbf{F}(\mathbf{x}(t+s),\mathbf{x}^{\prime}(t+s),t+s)=\mathbf{F}(\mathbf{x}(t+s),\mathbf{x}^{\prime}(t+s),t).$

The initial positions and initial velocities of points can be chosen in any
way and at any moment, so

$\mathbf{F}(\mathbf{x},\mathbf{y},t+s)=F(\mathbf{x},\mathbf{y},t)$

for all $\mathbf{x}\in\mathbb{R}^{N}$, $\mathbf{y}\in\mathbb{R}^{N}$,
$t\in\mathbb{R}$ and $s\in\mathbb{R}$. Thus the function $\mathbf{F}$ does not
depend on time and the equation of motion is of the form

$\mathbf{x}^{\prime\prime}(t)=F(\mathbf{x}(t),\mathbf{x}^{\prime}(t)).$ (7)

Fix now a vector $\mathbf{a}\in\mathbb{R}^{3}$. If $\mathbf{x}(t)$ satisfies
Eq. (4), then also the function
$\mathbf{y}(t)=(\mathbf{x}_{1}(t)+\mathbf{a},\ldots,\mathbf{x}_{n}(t)+\mathbf{a})$
satisfies Eq. (4). Thus, as before, we conclude that

$\mathbf{F}(\mathbf{x}_{1}+\mathbf{a},\ldots,\mathbf{x}_{n}+\mathbf{a},\mathbf{y})=\mathbf{F}(\mathbf{x}_{1},\ldots,\mathbf{x}_{n},\mathbf{y}),$

where $\mathbf{x}_{1}\in\mathbb{R}^{3},\ldots,\mathbf{x}_{n}\in\mathbb{R}^{3}$
and $\mathbf{y}\in\mathbb{R}^{N}$. From the invariance of the transformation
$g_{1}(t,\mathbf{x})=(t,\mathbf{x}+\mathbf{v}t)$ we conclude that

$\mathbf{F}(\mathbf{x},\mathbf{y}_{1}+\mathbf{v},\ldots,\mathbf{y}_{n}+\mathbf{v})=\mathbf{F}(\mathbf{x},\mathbf{y}_{1},\ldots,\mathbf{y}_{n})$

for all $\mathbf{x}\in\mathbb{R}^{N}$ and
$\mathbf{y}_{1}\in\mathbb{R}^{3},\ldots,\mathbf{y}_{n}\in\mathbb{R}^{3}$. If
$\mathbf{x}(t)$ is a solution of Eq. (4), then from invariance with respect to
$g_{3}(t,\mathbf{x})=(t,G\mathbf{x})$, the function
$\mathbf{y}(t)=(G\mathbf{x}_{1}(t),\ldots,G\mathbf{x}_{n}(t))$ is also a
solution of Eq. (4). Thus

$(G\mathbf{x}_{1}^{\prime\prime},\ldots,G\mathbf{x}_{n}^{\prime\prime})=\mathbf{F}(G\mathbf{x}_{1},\ldots,G\mathbf{x}_{n},G\mathbf{x}_{1}^{\prime},\ldots,G\mathbf{x}_{n}^{\prime}).$
(8)

Since

$(G\mathbf{x}_{1}^{\prime\prime},\ldots,G\mathbf{x}_{n}^{\prime\prime})=(G\mathbf{F}_{1}(\mathbf{x},\mathbf{x}^{\prime}),\ldots,G\mathbf{F}_{n}(\mathbf{x},\mathbf{x}^{\prime})),$

we obtain from Eq. (8) that

$\mathbf{F}(G\mathbf{x}_{1},\ldots,G\mathbf{x}_{n},G\mathbf{y}_{1},\ldots,G\mathbf{y}_{n})=(G\mathbf{F}_{1}(\mathbf{x},\mathbf{y}),\ldots,G\mathbf{F}_{n}(\mathbf{x},\mathbf{y}))$

for all
$\mathbf{x}_{1}\in\mathbb{R}^{3},\ldots,\mathbf{x}_{n}\in\mathbb{R}^{3}$,
$\mathbf{y}_{1}\in\mathbb{R}^{3},\ldots,\mathbf{y}_{n}\in\mathbb{R}^{3}$. Let
us introduce the following notations. If $\mathbf{a}\in\mathbb{R}^{3}$ and
$\mathbf{x}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})\in\mathbb{R}^{N}$, then
$\mathbf{x}+\mathbf{a}:=(\mathbf{x}_{1}+\mathbf{a},\ldots,\mathbf{x}_{n}+\mathbf{a})$
and if $G:\mathbb{R}^{3}\to\mathbb{R}^{3}$, then
$G\mathbf{x}:=(G\mathbf{x}_{1},\ldots,G\mathbf{x}_{n})$. Then the function
$\mathbf{F}$ satisfies the following conditions

  1. (i)

$\mathbf{F}(\mathbf{x},\mathbf{y})=\mathbf{F}(\mathbf{x}+\mathbf{a},\mathbf{y}+\mathbf{b})$,

  2. (ii)

$G\mathbf{F}(\mathbf{x},\mathbf{y})=\mathbf{F}(G\mathbf{x},G\mathbf{y})$

for all vectors $\mathbf{a}\in\mathbb{R}^{3},\mathbf{b}\in\mathbb{R}^{3}$, and
for all orthogonal transformations $G:\mathbb{R}^{3}\to\mathbb{R}^{3}$.

We now give an example of the application of the obtained result. Let us
consider a system consisting of only one point. Then it follows from equation
(i) that there exists a vector $\mathbf{c}\in\mathbb{R}^{3}$ such that
$\mathbf{F}(\mathbf{x},\mathbf{y})=\mathbf{c}$ for each
$\mathbf{x}\in\mathbb{R}^{3}$ and $\mathbf{y}\in\mathbb{R}^{3}$. From (ii) it
follows that we have $G\mathbf{c}=\mathbf{c}$ for any rotation $G$. Hence
$\mathbf{c}=\mathbf{0}$. Thus, a point moves according to the equation

$\mathbf{x}^{\prime\prime}(t)=\mathbf{0}.$

It implies that $\mathbf{x}(t)=\mathbf{a}t+\mathbf{b}$ for some vectors
$\mathbf{a}\in\mathbb{R}^{3}$ and $\mathbf{b}\in\mathbb{R}^{3}$. Thus, a point
moves in a straight line at constant velocity. We have proved Newton’s first
law of motion: if no force acts on a material point, then the point remains at
rest or keeps moving in a straight line at constant speed.

###  2.5 Measure-preserving dynamical systems

Let $(X,\Sigma,\mu)$ be a probability space, i.e. $\mu$ is a probability
measure defined on a $\sigma$-algebra $\Sigma$ of subsets of $X$. Consider a
one-parameter semigroup $\\{\pi_{t}\\}_{t\in T}$. We assume that the function
$(t,x)\mapsto\pi_{t}x$ is measurable as a function from the Cartesian product
$T\times X$ to $X$ and that the measure $\mu$ is invariant with respect to
each transformation $\pi_{t}$, $t\in T$, i.e. $\mu(\pi_{t}^{-1}(A))=\mu(A)$
for any set $A\in\Sigma$. Then the quadruple $(X,\Sigma,\mu,\pi_{t})$ is
called a measure-preserving dynamical system. The triple $(X,\Sigma,\mu)$ is
called a phase space and the set $\\{\pi_{t}x\colon t\in T\\}$ is called
trajectory of a point $x\in X$. As in Sec. 2.1, we divide measure-preserving
dynamical systems into endomorphisms, automorphisms, flows, and semiflows.

Our goal is to study properties of measure-preserving dynamical systems:
ergodicity, mixing and exactness. A set $A$ is called invariant with respect
to a semigroup $\\{\pi_{t}\\}_{t\in T}$, if $\pi_{t}^{-1}(A)=A$ for any
$A\in\Sigma$ and $t\in T$. The family of invariant sets forms a
$\sigma$-algebra $\Sigma_{\mathrm{inv}}$. If $\sigma$-algebra
$\Sigma_{\mathrm{inv}}$ is trivial, i.e. it consists only of sets of measure
$\mu$ zero or one, then the measure-preserving dynamical system is said to be
ergodic.

A stronger property than ergodicity is mixing. A measure-preserving dynamical
system $(X,\Sigma,\mu,\pi_{t})$ is called mixing if

$\lim_{t\to\infty}\mu(A\cap\pi_{t}^{-1}(B))=\mu(A)\mu(B)\quad\text{for all
$A,B\in\Sigma$}.$ (9)

Identifying the measure $\mu$ with the probability P, one can formulate the
condition (9) as follows:

$\lim_{t\to\infty}{\rm P}(\pi_{t}(x)\in A|x\in B)={\rm P}(A)\quad\text{for all
$A,B\in\Sigma$ and ${\rm P}(B)>0$},$

which means that the trajectories of almost all points enter a set $A$ with
asymptotic probability $\mu(A)$.

A stronger property than mixing is exactness. A system
$(X,\Sigma,\mu,\pi_{t})$ with double measurable transformations $\pi_{t}$,
i.e. $\pi_{t}(A)\in\Sigma$ and $\pi_{t}^{-1}(A)\in\Sigma$ for all $A\in\Sigma$
and $t\in T$, and with an invariant probability measure $\mu$ is called exact
if for every set $A\in\Sigma$ with $\mu(A)>0$ we have
$\lim_{t\to\infty}\mu(\pi_{t}(A))=1$. Exactness cannot hold for flows and
automorphisms because in these cases

$\mu(\pi_{t}(A))=\mu(\pi_{t}^{-1}(\pi_{t}(A)))=\mu(A).$

Exactness is equivalent to the following condition: $\sigma$-algebra
$\bigcap_{t\geq 0}\pi_{t}^{-1}(\Sigma)$ contains only sets of measure $\mu$
zero or one. Here $\pi_{t}^{-1}(\Sigma)=\\{\pi_{t}^{-1}(A)\colon
A\in\Sigma\\}$.

Directly checking whether a dynamical system is ergodic, mixing or exact is
not an easy issue. This problem can be reduced to the study of linear
operators related to the representation of semigroups. In order to do it we
introduce the notion of Frobenius–Perron operators.

Let $(X,\Sigma,m)$ be a $\sigma$-finite measure space. A measurable map
$\varphi\colon X\to X$ is called non-singular if it satisfies the following
condition

$m(A)=0\Longrightarrow m(\varphi^{-1}(A))=0\textrm{ for $A\in\Sigma$.}$ (10)

Let $L^{1}=L^{1}(X,\Sigma,m)$ and let $\varphi$ be a measurable nonsingular
transformation of $X$. An operator $P_{\varphi}\colon L^{1}\to L^{1}$ which
satisfies the following condition

$\int_{A}P_{\varphi}f(x)\,m(dx)=\int_{\varphi^{-1}(A)}f(x)\,m(dx)\textrm{\,
for $A\in\Sigma$ and $f\in L^{1}$}$ (11)

is called the Frobenius–Perron operator for the transformation $\varphi$. The
operator $P_{\varphi}$ is linear, positive (if $f\geq 0$ then
$P_{\varphi}f\geq 0$) and preserves the integral
($\int_{X}P_{\varphi}f\,dm=\int_{X}f\,dm$). Any operator having the above
properties is called stochastic (or Markov) operator. We can equivalently
define a stochastic operator as a linear operator on $L^{1}$ carrying out
densities on densities. We recall that $f\in L^{1}$ is a density if $f\geq 0$
and $\|f\|=1$. By $D$ we denote the set of all densities.

The adjoint of the Frobenius–Perron operator $P^{*}\colon L^{\infty}\to
L^{\infty}$ is given by $P_{\varphi}^{*}g(x)=g(\varphi(x))$, thus
$P^{*}_{\varphi}$ is the Koopman operator restricted to the space
$L^{\infty}=L^{\infty}(X,\Sigma,m)$.

###### Example 1

Let $X$ be a subset of $\mathbb{R}^{d}$ with nonempty interior and with the
boundary of zero Lebesgue measure. Let $\varphi\colon X\to X$ be a measurable
transformation. We assume that there exists at most countable family of
pairwise disjoint open sets $U_{i}$, $i\in I$, contained in $X$ having the
following properties:

  1. a)

the sets $X_{0}=X\setminus\bigcup_{i\in I}U_{i}$ and $\varphi(X_{0})$ have
zero Lebesgue measure,

  2. b)

maps $\varphi_{i}=\varphi\Big{|}_{U_{i}}$ are diffeomorphisms from $U_{i}$
onto $\varphi(U_{i})$, i.e., $\varphi_{i}$ are $C^{1}$ and invertible
transformations and $\det\varphi^{\prime}_{i}(x)\neq 0$ at each point $x\in
U_{i}$.

Then transformations $\psi_{i}=\varphi_{i}^{-1}$ are also diffeomorphisms from
$\varphi(U_{i})$ onto $U_{i}$ and the Frobenius–Perron operator $P_{\varphi}$
exists and is given by the formula

$P_{\varphi}f(x)=\sum_{i\in I_{x}}f(\psi_{i}(x))|\det\psi_{i}^{\prime}(x)|,$
(12)

where $I_{x}=\\{i\colon x\in\varphi(U_{i})\\}$ (see Rudnicki-LN ; RT-K-k ).

###### Example 2

If we consider a flow $\\{\pi_{t}\\}_{t\in\mathbb{R}}$ on $X$ related to Eq.
(2), then according to formula (12) the group $\\{P^{t}\\}_{t\in\mathbb{R}}$
of Frobenius–Perron operators is given by the formula

$P^{t}f(x)=f(\pi_{-t}x)\det\Big{[}\dfrac{d}{dx}\pi_{-t}x\Big{]}.$

The set of all nonsingular transformations with composition as the binary
operation forms a monoid $(\mathbf{G},\circ)$. The set of all stochastic
operators ${\rm StO}(X)$ is also a monoid with composition as the binary
operation and ${\rm StO}(X)$ is a linear space. Since

$P_{\varphi\circ\psi}=P_{\varphi}\circ P_{\psi},$

the map $h\colon\mathbf{G}\to{\rm StO}(X)$ given by $h(\varphi)=P_{\varphi}$
is a representation of the semigroup $\mathbf{G}$ on the space
$L^{1}(X,\Sigma,m)$.

Frobenius–Perron operators can be successfully used to study ergodic
properties of dynamical systems LiM . Let $\varphi\colon X\to X$ be a
nonsingular transformation of some $\sigma$-finite measure space
$(X,\Sigma,m)$. First, we check when a probability measure $\mu$ absolutely
continuous with respect to $m$ is invariant with respect to $\varphi$. Let
$f^{*}$ be the density of $\mu$ with respect to $m$, i.e.
$\mu(A)=\int_{A}f^{*}\,dm$ for $A\in\Sigma$. The measure $\mu$ is invariant
with respect to $\varphi$ if and only if

$\int_{\varphi^{-1}(A)}f^{*}(x)\,m(dx)=\int_{A}f^{*}(x)\,m(dx)\textrm{\, for
$A\in\Sigma$.}$

From (11) it follows immediately that the measure $\mu$ is invariant with
respect to $\varphi$ if and only if $P_{\varphi}f^{*}=f^{*}$. It means that if
the map $\pi\colon T\times X\to X$ is measurable and $\\{\pi_{t}\\}_{t\in T}$
is a one-parameter semigroup of nonsingular transformations of $(X,\Sigma,m)$,
$P^{t}$ denotes the Frobenius–Perron operator corresponding to $\pi_{t}$, then
the quadruple $(X,\Sigma,\mu,\pi_{t})$ is a measure-preserving dynamical
system if and only if $P^{t}f^{*}=f^{*}$ for all $t\in T$.

It is also not difficult to check that the measure $\mu$ is ergodic if and
only if $f^{*}$ is a unique fixed point of all operators $P^{t}$ in the set of
densities; and the dynamical system is mixing if and only if for every $f\in
D$ the density $f^{*}$ is the weak limit of $P^{t}f$ as $t\to\infty$. By the
weak limit $\lim_{t\to\infty}P^{t}f$ we understand a function $h\in L^{1}$
such that for every $g\in L^{\infty}$ we have

$\lim_{t\to\infty}\int_{X}P^{t}f(x)g(x)\,m(dx)=\int_{X}h(x)g(x)\,m(dx)$

and we denote it by w-$\lim_{t\to\infty}P^{t}f$. More difficult is to prove
that the dynamical system is exact if and only if
$\lim_{t\to\infty}P^{t}f=f^{*}$ for every $f\in D$.

We collect the relations between ergodic properties of a dynamical system and
the behavior of the Frobenius–Perron operator in Table 2.

$\mu$ | $f^{*}$
---|---
invariant |  $P^{t}f^{*}=f^{*}$ for all $t\in T$
ergodic |  $f^{*}$ is a unique fixed point in $D$ of all $P^{t}$
mixing | w-$\lim_{t\to\infty}P^{t}f=f^{*}$ for every $f\in D$
exact |  $\lim_{t\to\infty}P^{t}f=f^{*}$ for every $f\in D$
Table 2: The relations between ergodic properties of the dynamical system
$(X,\Sigma,\mu,\pi_{t})$ and the Frobenius–Perron operators $P^{t}$.

###### Example 3

Let $\varphi\colon[0,1]\to[0,1]$ be the transformation given by

$\varphi(x)=\begin{cases}2x&\textrm{for $x\in[0,1/2]$},\\\ 2-2x&\textrm{for
$x\in(1/2,1]$.}\end{cases}$ (13)

The transformation $\varphi$ is called the tent map because its graph looks
like a tent. The Frobenius–Perron operator $P_{\varphi}$ is of the form

$P_{\varphi}f(x)=\tfrac{1}{2}f(\tfrac{1}{2})+\tfrac{1}{2}f(1-\tfrac{1}{2}x).$
(14)

It is easy to see that the density $f^{*}=1_{[0,1]}$ satisfies
$P_{\varphi}f^{*}=f^{*}$, and consequently $\varphi$ is an endomorphism on the
phase space $([0,1],\mathcal{B}([0,1]),\ell)$, where $\mathcal{B}([0,1])$ is
the $\sigma$-algebra of Borel subsets of $[0,1]$ and $\ell$ is the Lebesgue
measure. We check that $\lim_{t\to\infty}P_{\varphi}^{t}f=f^{*}$ for any
density $f$. It is sufficient to check this condition for densities which are
Lipschitz continuous. Let $L$ be the Lipschitz constant for $f$. Then

$|P_{\varphi}f(x)-P_{\varphi}f(y)|\leq\tfrac{1}{2}|f(\tfrac{x}{2})-f(\tfrac{y}{2})|+\tfrac{1}{2}|f(1-\tfrac{1}{2}x)+f(1-\tfrac{1}{2}y)|\leq\tfrac{L}{2}|x-y|.$

Thus $L/2$ is the Lipschitz constant for $P_{\varphi}f$ and by induction we
conclude that $L/2^{t}$ is the Lipschitz constant for $P_{\varphi}^{t}f$.
Hence, the sequence $(P_{\varphi}^{t}f)$ converges uniformly to a constant
function. Since $P_{\varphi}^{t}f$ are densities, $(P_{\varphi}^{t}f)$
converges to $f^{*}$ uniformly, which implies the convergence in $L^{1}$. The
condition $\lim_{t\to\infty}P_{\varphi}^{t}f=f^{*}$ implies the exactness of
the endomorphism $\varphi$.

###### Example 4

Now we check that the logistic map $\psi(x)=4x(1-x)$ on $[0,1]$ is exact. Let
$\varphi$ be the tent map and $\alpha(x)=\tfrac{1}{2}-\tfrac{1}{2}\cos(\pi
x)$. Then $\psi\circ\alpha=\alpha\circ\varphi$. This implies that
$P_{\psi}P_{\alpha}=P_{\alpha}P_{\varphi}$. Hence
$P_{\psi}=P_{\alpha}P_{\varphi}P_{\alpha^{-1}}$ and, by induction,
$P^{t}_{\psi}=P_{\alpha}P^{t}_{\varphi}P_{\alpha^{-1}}$. Let $f\in D$. Then
$P_{\alpha^{-1}}f\in D$ and from the previous example
$\lim_{t\to\infty}P^{t}_{\varphi}P_{\alpha^{-1}}f=1_{[0,1]}$. Therefore

$\lim_{t\to\infty}P^{t}_{\psi}f=P_{\alpha}1_{[0,1]}=g^{*},\quad\text{where
$g^{*}(x)=\Big{[}\pi\sqrt{x(1-x)}\,\Big{]}^{-1}$}.$

Thus the map $\psi$ is an exact endomorphism on the space
$([0,1],\mathcal{B}([0,1]),\mu)$, where $d\mu=g^{*}(x)\,dx$.

More advanced examples of applications of Frobenius–Perron operators to study
ergodic properties of dynamical systems can be found in BG ; LiM ; LY .

##  3 Stochastic semigroups

###  3.1 Semigroups of operators

Stochastic semigroups belong to a wider family of semigroups called
$C_{0}$-semigroups of operators, so we first need to introduce the notion of a
$C_{0}$-semigroup and to show its relation with evolution equations.

Let $(E,\|\cdot\|)$ be a Banach space and let $\\{T(t)\\}_{t\geq 0}$ be a
family of linear bounded operators on $E$. The family $\\{T(t)\\}_{t\geq 0}$
is called a semigroup of bounded operators if it satisfies the following
conditions:

  1. (a)

$T(0)=I$, i.e., $T(0)x=x$,

  2. (b)

$T(t+s)=T(t)T(s)\quad\textrm{for}\quad s,\,t\geq 0$.

If additionally

  1. (c)

for each $x\in E$ the function $t\mapsto T(t)x$ is continuous,

then $\\{T(t)\\}_{t\geq 0}$ is called a $C_{0}$-semigroup or strongly
continuous semigroup. Let $\\{T(t)\\}_{t\geq 0}$ be a $C_{0}$-semigroup and
let $\mathfrak{D}(A)$ be the set of such $x\in E$, that there exists the limit

$Ax=\lim_{t\to 0^{+}}\frac{T(t)x-x}{t}.$ (15)

Then the set $\mathfrak{D}(A)$ is a linear subspace dense in $E$, and $A$ is a
linear operator from $\mathfrak{D}(A)$ to $E$. The operator $A$ is called the
infinitesimal generator (briefly the generator) of the semigroup
$\\{T(t)\\}_{t\geq 0}$. We also say that the operator
$A\colon\mathfrak{D}(A)\to E$ generates the semigroup $\\{T(t)\\}_{t\geq 0}$.

The notion of a $C_{0}$-semigroup and its generator is strictly connected with
differential equations in the Banach space $E$. Let $\\{T(t)\\}_{t\geq 0}$ be
a $C_{0}$-semigroup on $E$ and let $A:\mathfrak{D}(A)\to E$ be its generator.
Then for every $x_{0}\in\mathfrak{D}(A)$ the function $x\colon[0,\infty)\to E$
defined by the formula $x(t)=T(t)x_{0}$ is differentiable (in the sense of
Fréchet) for $t\geq 0$ and satisfies the equation

$x^{\prime}(t)=Ax(t)\quad\textrm{with initial condition $x(0)=x_{0}$}.$ (16)

An equation of the form (16) is called an evolution equation. We also say that
equation (16) generates semigroup $\\{T(t)\\}_{t\geq 0}$.

In the theory of semigroups of operators, the key issue is when a linear
operator is a generator of a $C_{0}$-semigroup. If $A$ is a bounded linear
operator on $E$, then $A$ generates a $C_{0}$-semigroup on $E$ given by the
formula

$T(t)x=e^{At}x=\sum_{k=0}^{\infty}\frac{t^{k}A^{k}x}{k!}$ (17)

and the semigroup $\\{T(t)\\}_{t\geq 0}$ is uniformly continuous, i.e.,

$\lim_{t\to t_{0}}\|T(t)-T(t_{0})\|=0\textrm{ \,for $t_{0}\geq 0$}.$ (18)

We will study mainly semigroups of contractions, so we only characterize the
generators of such semigroups. We recall that a linear operator $T$ on $E$ is
a contraction if $\|Tx\|\leq\|x\|$ for $x\in E$.

######  Theorem 3.1 (Hille–Yosida)

Let $E$ be a Banach space, and let $A$ be a linear operator on $E$ with domain
$\mathfrak{D}(A)$. The operator $A$ is a generator of a strongly continuous
semigroup of contractions if and only if the following conditions are
satisfied:

  1. (a)

the set $\mathfrak{D}(A)$ is dense in $E$;

  2. (b)

for each $\lambda>0$ there exists an operator $(\lambda I-A)^{-1}$ defined
everywhere on $E$;

  3. (c)

$\|(\lambda I-A)^{-1}\|\leq\lambda^{-1}$ for each $\lambda>0$.

The operator $(\lambda I-A)^{-1}$ is called the resolvent of $A$ at $\lambda$
and it is denoted by $R(\lambda,A)$ or $R_{\lambda}$. The resolvent is defined
for $\lambda$ from the resolvent set $\rho(A)$ which consists of all
$\lambda\in\mathbb{C}$ such that the operator $\lambda I-A$ is invertible and
the inverse operator is a bounded operator defined on the whole space $E$. The
set $\sigma(A)=\mathbb{C}\setminus\rho(A)$ is called the spectrum of the
operator $A$. If the operator $A$ generates a semigroup $\\{T(t)\\}_{t\geq
0}$, then $R_{\lambda}$ is called the resolvent of the semigroup
$\\{T(t)\\}_{t\geq 0}$. The resolvent of a strongly continuous semigroup of
contractions is also given by the formula

$R_{\lambda}x=\int_{0}^{\infty}e^{-\lambda t}T(t)x\,\,dt\quad\ \text{for $x\in
E$}.$ (19)

###### Remark 1

We can also consider nonlinear strongly continuous semigroups. A family
$\\{T(t)\\}_{t\geq 0}$ of nonlinear operators on a Banach space is called a
nonlinear strongly continuous semigroup if it satisfies conditions (a), (b),
and the map $(t,x)\mapsto T(t)x$ is continuous.

###  3.2 Stochastic and substochastic semigroups

Let $(X,\Sigma,m)$ be a $\sigma$-finite measure space and let
$L^{1}=L^{1}(X,\Sigma,m)$. A $C_{0}$-semigroup $\\{P(t)\\}_{t\geq 0}$ of
positive contractions on $L^{1}$ is called a substochastic semigroup. A
$C_{0}$-semigroup of stochastic operators is called a stochastic semigroup or
a Markov semigroup. We can also consider a nonlinear stochastic semigroup if
$\\{P(t)\\}_{t\geq 0}$ is a nonlinear strongly continuous semigroup and
$P(t)(D)\subset D$ for $t\geq 0$, where $D$ is the set of all densities.

An example of a stochastic semigroup is a semiflow of Frobenius–Perron
operators $\\{P^{t}\\}_{t\geq 0}$ considered in Sec. 2.5 if we assume that
this semiflow satisfies condition (c). We will now examine what conditions a
linear operator $A$ on a $L^{1}$ space should satisfy in order to be a
generator of a stochastic semigroup.

First we consider the case when $A$ is a bounded operator. A bounded linear
operator $A$ is a generator of a positive semigroup on $L^{1}$ (generally on a
Banach lattice) if and only if $A+\|A\|I\geq 0$ (see e.g. Theorem 1.11, page
255 Arendt ). We recall that a semigroup $\\{P(t)\\}_{t\geq 0}$ is positive if
each operator $P(t)$ is positive. Let $\lambda=\|A\|$ and $B=A+\lambda I$,
then $B$ is a positive operator. Thus the generator of the semigroup of
positive operators $\\{P(t)\\}_{t\geq 0}$ on $L^{1}$ is of the form
$A=-\lambda I+B$, where $B$ is a positive and bounded operator. If $A$ is a
generator of a stochastic semigroup then $P(t)$ preserves the integral. Since
$P(t)=I+tA+o(t)$, we obtain

$\int_{X}Af(x)\,m(dx)=0\quad\textrm{for $f\in L^{1}$}.$ (20)

Thus $\int_{X}Bf(x)\,m(dx)=\lambda$ for $f\in D$. Assume that $\lambda>0$ and
let $P=B/\lambda$. Then $A=-\lambda I+\lambda P$, where $P$ is a stochastic
operator. Analogously if a linear and bounded operator $A$ is a generator of a
substochastic semigroup, then $A$ is of the form $A=-\lambda I+\lambda P$,
where $P$ is a substochastic operator and $\lambda\geq 0$. The semigroup
$\\{P(t)\\}_{t\geq 0}$ is given by the formula

$P(t)f=\sum_{k=0}^{\infty}\frac{(\lambda t)^{k}e^{-\lambda t}}{k!}P^{k}f.$
(21)

###### Example 5

Let $X=\\{1,2,\dots,n\\}$, $\Sigma=2^{X}$ and let $m$ be the counting measure.
In this case the space $L^{1}(X,\Sigma,m)$ is denoted by $l^{1}_{n}$. In the
space $l^{1}_{n}$ a function $f\colon X\to\mathbb{R}$ is represented as a
sequence $y=(y_{1},\dots,y_{n})$, the integral of $y$ over $X$ is given by
$\sum_{i=1}^{n}y_{i}$, and the space $l^{1}_{n}$ is isomorphic to the space
$\mathbb{R}^{n}$ with the norm $\|y\|=|y_{1}|+\dots+|y_{n}|$. The generator
$A$ of a stochastic semigroup on $l^{1}_{n}$ is of the form $Ay=yQ$, where the
matrix $Q=[q_{ij}]$ satisfies conditions

  1. (i)

$q_{ij}\geq 0$ for $i\not=j$,

  2. (ii)

$\sum_{j=1}^{n}q_{ij}=0$ for $i=1,\dots,n$.

The matrix $Q$ is the intensity matrix of a Markov chain $(\xi_{t})_{t\geq
0}$. The entry $q_{ij}$, $i\neq j$, is the rate of jump from the state $i$ to
$j$, precisely

${\rm P}(\xi_{t+\Delta t}=j|\xi_{t}=i)=q_{ij}\Delta t+o(\Delta t).$ (22)

Moreover, $|q_{ii}|$ is the rate of leaving the state $i$.

###### Remark 2

One can ask the question whether condition (20) is sufficient for a bounded
linear operator to generate a stochastic semigroup. The answer is negative.
Let $X=[0,1]$ and $m$ be the Lebesgue measure. Consider the operator $A=I-P$,
where $Pf(x)=2f(2x)$ for $x\in[0,1/2]$ and $Pf(x)=0$ otherwise. The operator
$A$ satisfies (20) and generates a semigroup $\\{T(t)\\}_{t\geq 0}$ but this
semigroup is not positive. We have

$T(t)f=e^{t}e^{-tP}f=e^{t}\sum_{k=0}^{\infty}\frac{(-1)^{k}t^{k}P^{k}}{k!}f.$

If $f=\mathbf{1}_{(1/2,1]}$, then
$P^{k}f=2^{k}\mathbf{1}_{(1/2^{k+1},1/2^{k}]}$ and $T(t)f(x)<0$ for
$x\in(1/2^{k+1},1/2^{k}]$ and $k$ odd.

Theorem 3.1 can be applied to check whether an operator $A$ is a generator of
a stochastic or substochastic semigroup. From (19) it follows that if a
semigroup is substochastic, then the operators $R_{\lambda}$ are positive for
each $\lambda>0$. It turns out that vice versa, too, if $A$ satisfies
conditions (a)–(c) of Theorem 3.1 and

  1. (d)

$R_{\lambda}f\geq 0$ for all $\lambda>0$ and $f\geq 0$,

then $A$ is the generator of a substochastic semigroup.

If a semigroup $\\{P(t)\\}_{t\geq 0}$ is stochastic, then from (19) it follows
that the operators $\lambda R_{\lambda}$ are stochastic. If the operator $A$
satisfies conditions (a)–(d) and additionally

  1. (e)

for some $\lambda>0$ the operator $\lambda R_{\lambda}$ is stochastic,

then $A$ generates a stochastic semigroup. Condition (e) can be replaced by
the following:

  1. (e′)

$\int_{X}Af(x)\,m(dx)=0$ for $f\in\mathfrak{D}(A)$.

In applications, the following problem often arises. We know that an operator
$A$ is the generator of a stochastic or substochastic semigroup
$\\{P(t)\\}_{t\geq 0}$ and $B$ is a linear operator on $L^{1}$. When the
operator $A+B$ also generates a stochastic or a substochastic semigroup? This
question is answered by perturbation theorems. Now we present one of them
useful in applications.

###### Theorem 3.2

Let $A$ be a generator of a stochastic semigroup $\\{S(t)\\}_{t\geq 0}$, $K$
be a stochastic operator and $\lambda>0$. The semigroup $\\{P(t)\\}_{t\geq 0}$
generated by the operator $A+\lambda K-\lambda I$ is stochastic and it is
given by the Dyson–Phillips expansion:

$P(t)f=e^{-\lambda t}\sum_{n=0}^{\infty}\lambda^{n}S_{n}(t)f,$ (23)

where

$S_{0}(t)f=S(t)f,\quad S_{n+1}(t)f=\int_{0}^{t}S_{n}(t-s)KS(s)f\,ds,\quad
n\geq 0.$ (24)

If $\\{S(t)\\}_{t\geq 0}$ is a substochastic semigroup and $K$ is a
substochastic operator then $\\{P(t)\\}_{t\geq 0}$ is a substochastic
semigroup.

###  3.3 Semigroups related to Markov chains

It can be expected that the construction of the stochastic semigroup presented
in Example 5 can be extended to the case where $X$ is a countable set.
However, this case is more difficult.

Let $l^{1}=L^{1}(\mathbb{N},2^{\mathbb{N}},m)$, where $m$ is the counting
measure. Now elements of $l^{1}$ are infinite sequences
$y=(y_{0},y_{1},\dots)$ and $\|y\|=\sum_{i=0}^{\infty}|y_{i}|$. We assume that
an infinity dimensional matrix $Q=[q_{ij}]$ satisfies conditions

  1. (i)

$q_{ij}\geq 0$ for $i\not=j$,

  2. (ii)

$\sum_{j=0}^{\infty}q_{ij}=0$ for $i=0,1,\dots$.

If $\sup_{i\in\mathbb{N}}|q_{ii}|<\infty$, then the operator $Ay=yQ$ is
bounded and $P(t)y=e^{At}y$, $t\geq 0$, is a stochastic semigroup on $l^{1}$.

The case when $A$ is an unbounded operator is more complex. Two linear
subspaces can be associated with the matrix $Q$:

$\mathfrak{D}_{0}(Q)=\\{x\in
l^{1}\colon\sum_{i=0}^{\infty}|q_{ii}||x_{i}|<\infty\\},$
$\displaystyle\mathfrak{D}(Q)=\bigg{\\{}x\in l^{1}\colon$ for each
$j\in\mathbb{N}$ the series $\sum_{i=0}^{\infty}x_{i}q_{ij}$ is absolutely
$\displaystyle\text{ convergent and
\,}\sum_{j=0}^{\infty}\bigg{|}\sum_{i=0}^{\infty}x_{i}q_{ij}\bigg{|}<\infty\bigg{\\}}.$

Firstly, a mapping $A$ with domain $\mathfrak{D}_{0}(Q)$ or $\mathfrak{D}(Q)$
need not be the generator of a strongly continuous semigroup on $l^{1}$.
Secondly, even when $A$ with an appropriately chosen domain generates a
strongly continuous semigroup on $l^{1}$, this semigroup may not be a
stochastic semigroup. Thirdly, the same matrix $Q$ can be used to generate
different positive semigroups (including stochastic semigroups) on $l^{1}$
depending on the choice of the domain of $A$.

We consider a family $\mathfrak{F}$ of all positive semigroups on $l^{1}$ with
generators $A$ such that: $\mathfrak{D}_{0}(Q)\subseteq\mathfrak{D}(A)$ and
$Ax=xQ$ for $x\in\mathfrak{D}_{0}(Q)$. A semigroup $\\{P(t)\\}_{t\geq 0}$ from
$\mathfrak{F}$ is called minimal if for any other semigroup $\\{T(t)\\}_{t\geq
0}$ from $\mathfrak{F}$ we have $T(t)f\geq P(t)f$ for each nonnegative
$f\in\mathfrak{D}_{0}(Q)$ and $t\geq 0$.

###### Theorem 3.3

Assume that the matrix $Q$ satisfies (i)–(ii). Then there is the minimal
semigroup $\\{P(t)\\}_{t\geq 0}$ and $\\{P(t)\\}_{t\geq 0}$ is a substochastic
semigroup.

The answer for the question when a minimal semigroup is stochastic is given by
following Kato theorem kato54 .

###### Theorem 3.4

Let $\lambda>0$ be a positive constant. The minimal semigroup related to $Q$
is a stochastic semigroup if and only if the equation $Qx=\lambda x$ has no
nonzero and nonnegative solution $x\in l^{\infty}$.

If the minimal semigroup $\\{P(t)\\}_{t\geq 0}$ related to a matrix $Q$ is a
stochastic semigroup, then the matrix $Q$ and the semigroup $\\{P(t)\\}_{t\geq
0}$ are called non-explosive. If the matrix is non-explosive, then $Q$ is an
intensity matrix of a Markov chain $(\xi_{t})_{t\geq 0}$ on $\mathbb{N}$, i.e.
the entries $q_{ij}$ satisfy condition (22).

###### Remark 3

If $\mathfrak{D}(A)$ is the domain of the generator of a stochastic semigroup
related to the intensity matrix $Q$, then

$\mathfrak{D}_{0}(Q)\subseteq\mathfrak{D}(A)\subseteq\mathfrak{D}(Q)$

and none of these inclusions can be replaced by equality.

###  3.4 Diffusion semigroup

The theory of stochastic processes provides many interesting examples of
semigroups. Semigroups related to diffusion processes play a fundamental role.

Consider the Itô equation

$dx_{t}=\sigma(x_{t})\,dw_{t}+b(x_{t})\,dt,$ (25)

where $w_{t}$ is an $m$-dimensional Brownian motion,
$\sigma(x)=[\sigma_{j}^{i}(x)]$ is a $d\times m$ matrix and $b(x)$ is a vector
in $\mathbb{R}^{d}$ for every $x\in\mathbb{R}^{d}$. We assume that for all
$i=1,...,d$, $j=1,...,m$ the functions $b_{i}$, $\sigma_{j}^{i}$ are
sufficiently smooth and have bounded derivatives of all orders, and the
functions $\sigma_{j}^{i}$ are also bounded. A process $x_{t}$, $t\geq 0$,
which satisfies (25) is called diffusion process.

Assume that the random variable $x_{0}$ has the density $f(x)$. Then solution
$x_{t}$ of (25) also has the density $u(t,x)$ and $u$ satisfies the
Fokker–Planck equation (also called the Kolmogorov forward equation)

$\frac{\partial u}{\partial
t}=\sum_{i,j=1}^{d}\frac{\partial^{2}(a_{ij}(x)u)}{\partial x_{i}\partial
x_{j}}-\sum_{i=1}^{d}\frac{\partial(b_{i}(x)u)}{\partial x_{i}},$ (26)

where $a_{ij}(x)=\frac{1}{2}\sum_{k=1}^{m}\sigma^{i}_{k}(x)\sigma^{j}_{k}(x)$.
Let $P(t)f(x)=u(t,x)$. Then $\\{P(t)\\}_{t\geq 0}$ is a stochastic semigroup
(Rudnicki-LN ).

Consider the operator

$A_{0}f=\sum_{i,j=1}^{d}\frac{\partial^{2}(a_{ij}(x)f)}{\partial x_{i}\partial
x_{j}}-\sum_{i=1}^{d}\frac{\partial(b_{i}(x)f)}{\partial x_{i}}$ (27)

on the set $E=\\{f\in L^{1}(\mathbb{R}^{d})\cap
C^{2}_{b}(\mathbb{R}^{d})\colon A_{0}f\in L^{1}(\mathbb{R}^{d})\\}$, where
$C^{2}_{b}(\mathbb{R}^{d})$ denotes the set of all twice differentiable
bounded functions whose derivatives of order ${}\leq 2$ are continuous and
bounded. The closure of the operator $A_{0}$ is the generator of a stochastic
semigroup $\\{P(t)\\}_{t\geq 0}$ ((EK, , Proposition 1.3.3)). We recall that
the closure of the operator $\,A_{0}\,$ is the operator $A$ whose graph is the
closure of the graph of $A_{0}$, as long as such an operator $A$ exists.

If the functions $a_{ij}$ satisfy the uniform elliptic condition:

$\sum_{i,j=1}^{d}a_{ij}(x)\lambda_{i}\lambda_{j}\geq\alpha\|\lambda\|^{2}$
(28)

for some $\alpha>0$ and every $\lambda\in\mathbb{R}^{d}$ and
$x\in\mathbb{R}^{d}$, then the stochastic semigroup $\\{P(t)\\}_{t\geq 0}$ is
an integral semigroup, i.e.

$P(t)f(y)=\int_{\mathbb{R}^{d}}q(t,x,y)f(x)\,dx,\quad t>0,$

and the kernel $q$ is continuous and positive.

It should be mentioned that in the case when $\sigma$ is a null matrix, then
Eq. (25) reduces to Eq. (2). Consequently, the generator of the group of
Frobenius–Perron operators from Example 2 is of the form

$Af=-\sum_{i=1}^{d}\frac{\partial(b_{i}(x)f)}{\partial x_{i}}.$ (29)

###  3.5 Processes with jumps

Let $(X,\Sigma,m)$ be a $\sigma$-finite measure space and let $(x_{t})_{t\geq
0}$ be a family of Markov processes such that if the distribution of the
random variable $x_{0}$ has a density $f$, then the random variable $x_{t}$
has a density $S(t)f$ and $\\{S(t)\\}_{t\geq 0}$ is a stochastic semigroup.
Let $(A_{0},\mathfrak{D}(A_{0}))$ be the generator of the semigroup
$\\{S(t)\\}_{t\geq 0}$. For example $\\{S(t)\\}_{t\geq 0}$ can be a diffusion
semigroup or a semigroup of Frobenius–Perron operators.

Consider a family of Markov processes $(\xi_{t})_{t\geq 0}$ describing the
motion of points in the space $X$ and defined in the following way. A point
moves along the sample paths of the process $(x_{t})_{t\geq 0}$ but it can
randomly jump to a new position according to a jump function defined below and
then it continues to move along the sample paths of the process
$(x_{t})_{t\geq 0}$, etc.

We define the jump function by specifying the jump intensity from a point $x$
to a set $B\in\Sigma$. For a fixed point $x\in X$, the jump function is a
measure $\mathcal{K}(x,B)$ defined in such a way that the probability of jump
from a point $x$ to a set $B$ during the time $\Delta t$ is
$\mathcal{K}(x,B)\Delta t+o(\Delta t)$. Usually the jump function
$\mathcal{K}(x,B)$ is of the form $\mathcal{K}(x,B)=\psi(x)\mathcal{P}(x,B)$,
where $\psi(x)$ is the rate of jump and $\mathcal{P}(x,B)$ is the probability
of jump from $x$ to a given set $B$. The function $\mathcal{P}$ is a
transition probability function on $(X,\Sigma)$, i.e. $\mathcal{P}(x,\cdot)$
is a probability measure on $(X,\Sigma)$ for each $x\in X$ and the function
$x\mapsto\mathcal{P}(\cdot,B)$ is measurable for each $B\in\Sigma$.

Assume that $\mathcal{P}$ has the following property

$m(B)=0\Longrightarrow\mathcal{P}(x,B)=0\textrm{ for $\,m$-a.e. $x\,$ and
$B\in\Sigma$}.$ (30)

Then for every $f\in D$ the measure

$\mu(B)=\int f(x)\mathcal{P}(x,B)\,m(dx)$

is absolutely continuous with respect to the measure $m$. This fact is a
simple consequence of the Radon–Nikodym theorem, which says that the measure
$\nu$ is absolutely continuous with respect to the measure $m$ if and only if
the following implication $m(B)=0\Rightarrow\nu(B)=0$ holds for all sets
$B\in\Sigma$. Now, the formula $Pf=d\mu/dm$ defines a stochastic operator
$P:L^{1}\to L^{1}$ called a jump operator. Moreover, if $P^{*}\colon
L^{\infty}\to L^{\infty}$ is the adjoint operator of $P$, then $P^{*}g(x)=\int
g(y)\,\mathcal{P}(x,dy)$.

We consider two examples of the jump operator. One of them is Frobenius–Perron
operator $P_{\varphi}$ introduced in Sec. 2.5. The transition probability
function corresponding to $P_{\varphi}$ is of the form

$\mathcal{P}(x,B)=\begin{cases}1,\textrm{ if $\varphi(x)\in B$,}\\\ 0,\textrm{
if $\varphi(x)\notin B$.}\end{cases}$ (31)

The second example is an integral stochastic operator. If $(X,\Sigma,m)$ is a
$\sigma$-finite measure space $q\colon X\times X\to[0,\infty)$ is a measurable
function such that

$\int_{X}q(x,y)\,m(dy)=1$

for almost all $x\in X$, then

$Pf(y)=\int_{X}q(x,y)f(x)\,m(dx)$ (32)

is a stochastic operator. The function $q$ is called the kernel of the
operator $P$ and $P$ is called an integral or kernel operator. The function
$\mathcal{P}(x,B)=\int_{B}q(x,y)\,m(dy)$ is the transition probability
function corresponding to $P$.

We assume that the function $\psi\colon X\to[0,\infty)$ describing the
intensity of jumps is measurable and bounded. Then the process
$(\xi_{t})_{t\geq 0}$ defines a semigroup $\\{P(t)\\}_{t\geq 0}$ on $L^{1}$
with generator

$Af=A_{0}f-\psi f+P(\psi f).$ (33)

We check that $\\{P(t)\\}_{t\geq 0}$ is a stochastic semigroup. Let
$\lambda=\sup_{X}\psi(x)$ and $Kf=\lambda^{-1}[P(\psi f)+(\lambda-\psi)f]$.
Then $K$ is a stochastic operator and $A=A_{0}-\lambda I+\lambda K$. According
to Theorem 3.2 the operator $(A,\mathfrak{D}(A_{0}))$ is the generator of a
stochastic semigroup defined by formulas (23) and (24).

If $A_{0}$ is of the form (29) and is therefore the generator of a semigroup
of Frobenius–Perron operators, then $(\xi_{t})_{t\geq 0}$ is an example of a
piecewise deterministic process davis84 . A stochastic process
$(\xi_{t})_{t\geq 0}$ is called a piecewise deterministic processes if there
is an increasing sequence of random times $(t_{n})$, called jumps, such that
sample paths of $(\xi_{t})_{t\geq 0}$ are defined in a deterministic way in
each interval $[t_{n},t_{n+1})$; in our case they satisfy Eq. (2).

###  3.6 Diffusion process with random switching

Consider the following stochastic equation:

$dx_{t}=\sigma(x_{t},y_{t})\,dw_{t}+b(x_{t},y_{t})\,dt,$ (34)

where the process $(y_{t})_{t\geq 0}$ takes values in the set
$I=\\{0,1,\dots,n\\}$, $n\geq 1$. We assume that the matrices $\sigma(x,k)$
and the vectors $b(x,k)$ satisfy the same assumptions as $\sigma(x)$ and
$b(x)$ from Sec. 3.4. We also assume that if $y_{t}=k$, then the process
$(y_{t})_{t\geq 0}$ jumps to the state $l$ with rate $q_{kl}(x_{t})$, i.e.

${\rm P}(y_{t+\Delta t}=l|y_{t}=k)=q_{kl}(x_{t})\Delta t+o(\Delta t).$ (35)

We assume that the functions $q_{kl}$ are bounded, continuous and nonnegative.
Then $\xi_{t}=(x_{t},y_{t})$, $t\geq 0$ is a Markov process with values in the
space $X=\mathbb{R}^{d}\times I$ and it is called diffusion process with
random switching Y-Zhu . Fig. 3 shows an example of the trajectory of such a
process.

Figure 3: Example of a trajectory of a diffusion process with random switches.
In this case $X=\mathbb{R}^{2}\times\\{0,1\\}$ and $t_{0},t_{1},t_{2},t_{3}$
are the successive switching times.

If $I=\\{0,1\\}$ and $\sigma(x,1)=0$, then Eq. (34) describes the process
which randomly jumps between stochastic and deterministic states. Such
processes appear in transport phenomena in sponge–type structures BBH ; Be ;
LR .

We will now find the stochastic semigroup related to the process
$(\xi_{t})_{t\geq 0}$. Let us denote by $\\{S^{k}(t)\\}_{t\geq 0}$ the
stochastic semigroup with diffusion matrix $\sigma(x,k)$ and with drift vector
$b(x,k)$. It has the generator of the form

$A_{k}f=\sum_{i,j=1}^{d}\frac{\partial^{2}(a_{ij}(x,k)f)}{\partial
x_{i}\partial x_{j}}-\sum_{i=1}^{d}\frac{\partial(b_{i}(x,k)f)}{\partial
x_{i}}.$ (36)

Let $q_{kk}(x)=-\sum_{l\neq k}q_{kl}(x)$. For a function $f$ of the variables
$(x,k)$ we use the notation $f=[f_{0},\dots,f_{n}]$, where $f_{k}(x)=f(x,k)$.
Let $Af=[A_{0}f_{0},\dots,A_{n}f_{n}]$ and $Qf=[f_{0},\dots,f_{n}]Q$. Then the
operator $A+Q$ is the generator of a stochastic semigroup $\\{P(t)\\}_{t\geq
0}$ associated with the process $(\xi_{t})_{t\geq 0}$. This semigroup is
defined on the space $E=L^{1}(X,\mathcal{B}(X),m)$, where $m$ is the product
of the Lebesgue measure on $\mathbb{R}^{d}$ and the counting measure on $I$.
The operator $A$ is the generator of a stochastic semigroup $\\{S(t)\\}_{t\geq
0}$ on $E$ given by the formula

$S(t)f=(S^{0}(t)f_{0},\dots,S^{n}(t)f_{n}).$

Let $\lambda=\sup_{x\in\mathbb{R}^{d}}\max\,\\{-q_{00}(x),\dots,-q_{nn}(x)\\}$
and $K=\lambda^{-1}Q+I$. Then $K$ is a stochastic operator and $Q=-\lambda
I+\lambda K$. Since the generator of the semigroup $\\{P(t)\\}_{t\geq 0}$ is
of the form $A-\lambda I+\lambda K$, this semigroup is given by the formulae
(23) and (24).

In the case when $\sigma\equiv 0$, the Markov process $(\xi_{t})_{t\geq 0}$ is
a piecewise deterministic. This process describes the action of several
dynamical systems with random switching between them. Such dynamical systems
have many applications in biology. They are natural models for describing gene
regulatory networks. Depending on the state of activity of the genes present
in the network, it is described by different systems of differential
equations. A change in gene activity switches the system BLPR ; lipniacki2 ;
RT ; SH ; ZFL ; ZFML .

###  3.7 Processes with non-random moments of jumps

We will now present some processes in which a change in dynamics or a jump in
phase space occurs when the process reaches the boundary of the domain or some
fixed subset of the phase space. An example of such a process is stochastic
billiards CPSV ; Evans-b ; LM-KR ; M-KR . Similar properties have biological
models of cell cycle LeR ; PR-cell-cycle ; PR-cykl-kom-prz ; Rotenberg ,
neuron activity Burkitt ; PR-Stein ; Stein1967 ; SGJ , gene regulatory
networks HWS ; KKLL , and vesicular transport in the cell BN1 ; BN2 ; BN3 .

Let $\Omega\subset\mathbb{R}^{d}$ be a compact set with nonempty interior and
with piecewise smooth boundary $\partial\Omega$. In stochastic billiards a
moving point does not change its velocity in the interior of $\Omega$, but
when it collides with its boundary $\partial\Omega$. The new velocity is
chosen randomly from among the velocities directed towards the interior of the
set of $\Omega$ and the motion continues until the next collision with the
boundary (see Fig. 4).

Figure 4: Left: classical billiards – angles of incidence are equal to the
angles of reflection. Right: stochastic billiards. Regular and diffuse
reflection: $\mathcal{V}$ – the specular reflection, thin vectors – diffuse
reflection.

Let $x_{t}$ be the position of a moving particle and $v_{t}$ be its velocity.
Consider a stochastic process $\xi_{t}=(x_{t},v_{t})$, $t\geq 0$. This process
takes values in the space $X=\Omega\times V$, $V=\mathbb{R}^{d}$. We can also
consider stochastic billiards when a particle is moving with constant speed
$\|v\|=1$. Then $V=S^{d-1}$, where $S^{d-1}$ is a unit sphere in the space
$\mathbb{R}^{d}$. Assume that if the particle strikes the boundary at point
$x$ and with velocity $v$, then a new velocity $\bar{v}$ is chosen according
to the density distribution $k(x,v,\bar{v})$. Assume that the distribution of
the random variable $\xi_{0}$ has density $u_{0}\in
L^{1}(X,\mathcal{B}(X),\ell)$, where $\ell$ is the Lebesgue measure on $X$.
Then each random variable $\xi_{t}$ has density $u(t,x,v)$ and the function
$u$ satisfies the following initial-boundary problem LM-KR :

$\displaystyle\frac{\partial u}{\partial
t}+v\cdot\nabla_{x}u(t,x,v)=0,\quad\text{where \
}\nabla_{x}=\left[\dfrac{\partial}{\partial
x_{1}},\dots,\dfrac{\partial}{\partial x_{d}}\right],$ (37) $\displaystyle
u(0,x,v)=u_{0}(x,v),$ (38) $\displaystyle u_{|\Gamma_{-}}=H(u_{|\Gamma_{+}}),$
(39)

where

$\Gamma_{\pm}=\left\\{(x,v)\in\partial\Omega\times V;\ \pm v\cdot
n(x)>0\right\\},$

$n(x)$ is the outward unit normal at $x\in\partial\Omega$ and $H$ is a linear
boundary operator relating the outgoing and incoming fluxes
$u_{\mid\Gamma_{+}}$ and $u_{\mid\Gamma_{-}}$. The operator $H$ is linear and
bounded on the trace spaces

$L^{1}_{\pm}=L^{1}(\Gamma_{\pm}\,,\,|v\cdot n(x)|\pi(dx)\otimes
m_{V}(dv))=L^{1}(\Gamma_{\pm},d\mu_{\pm}(x,v)),$

where $\pi$ denotes the Lebesgue surface measure on $\partial\Omega$ and
$m_{V}$ is the Lebesgue measure on the space $V$. The operator $H\colon
L^{1}_{+}\to L^{1}_{-}$ is given by the formula

$H\varphi(x,\bar{v})=\int_{\Gamma_{+}(x)}k(x,v,\bar{v})\varphi(x,v)|v\cdot
n(x)|\,m_{V}(dv),$

where $\Gamma_{+}(x)=\\{v\in V\colon(x,v)\in\Gamma_{+}\\}$. Under some
technical assumptions the solutions of the system (37)–(39) generate a
stochastic semigroup on the space $L^{1}(X,\mathcal{B}(X),\ell)$ and this
semigroup has an interesting long-time behaviour (see LM-KR ).

One-dimensional version of stochastic billiards was studied in M-KR . This
version was applied in PR-cykl-kom-prz  to study Lebowitz and Rubinow model of
the cell cycle LeR . Some models of the cell cycle will be presented in more
detail in Sec. 5.8.

##  4 Long-time behaviour of semigroups of operators

###  4.1 The Perron–Frobenius theorem

A good starting point to study long-time behaviour of semigroups of operators
is a continuous version of the classical Perron–Frobenius theorem. To recall
it, let $E$ be the space $\mathbb{R}^{n}$ with the scalar product
$\langle\cdot,\cdot\rangle$. The zero vector in $E$ is denoted by
$\mathbf{0}$. If $x\in E$, then we write $x>\mathbf{0}$ when $x_{i}>0$ and
$x\geq\mathbf{0}$ when $x_{i}\geq 0$ for $i=1,2,\dots,n$. Let $Q=[q_{ij}]$ be
an $n\times n$ real matrix. Then the formula $T(t)x=xe^{tQ}$ defines a
semigroup $\\{T(t)\\}_{t\geq 0}$ on $E$. We assume that

  1. (P)

$q_{ij}\geq 0$ for $i\neq j$,

  2. (I)

for all $i$ and $j$ there exists a sequence $(i_{1},i_{2},\dots,i_{m})$ such
that $i_{1}=i$, $i_{m}=j$ and $q_{i_{k}i_{k+1}}>0\,$ for $k=1,\dots,m-1$.

Condition (P) guarantees that the semigroup $\\{T(t)\\}_{t\geq 0}$ is
positive, i.e. $T(t)x\geq\mathbf{0}$ for $x\geq\mathbf{0}$. From both
conditions (P) and (I) it follows that there are a constant $r\in\mathbb{R}$
and vectors $x^{*}>\mathbf{0}$ and $y^{*}>\mathbf{0}$ such that

$\lim_{t\to\infty}e^{-r\,t}T(t)x=x^{*}\,\langle y^{*},x\rangle\quad\textrm{for
$x\in E$}$ (40)

and the convergence in (40) is exponential. The irreducibility assumption (I)
implies that the limit set in (40) is one-dimensional. A semigroup
$\\{T(t)\\}_{t\geq 0}$ satisfying condition (40) is said to have asynchronous
exponential growth. It should be mentioned that one can find in literature,
e.g. Webb-a , a more general definition of asynchronous exponential growth,
where it is merely assumed that $e^{-r\,t}T(t)x$ converges to a nonzero finite
rank operator.

This theorem is a consequence of quite nontrivial properties of the matrix
$Q$. First of all, the number $r$ is a dominant eigenvalue of $Q$, which means
that any other eigenvalue $\lambda$ satisfies $\mathrm{Re\,}\lambda<r$. Also,
the general eigenspace associated to $r$ is one-dimensional, and $x^{*}$ and
$y^{*}$ are left and right eigenvectors corresponding to $r$, i.e.
$x^{*}Q=rx^{*}$ and $Qy^{*}=ry^{*}$ (it follows that
$T(t)x^{*}=e^{r\,t}x^{*}$). Finally, the functional $\alpha$ defined by
$\alpha(x)=\langle y^{*},x\rangle$ is positive, i.e. $\alpha(x)\geq 0$ for
$x\geq 0$.

If assumption (I) is dropped, the dominant eigenvalue $r\in\mathbb{R}$ still
exists, but (40) does not hold. More specifically, if the general eigenspace
associated to $r$ is one-dimensional then (P) implies $\eqref{c-d2}$ but the
inequalities $x^{*}>\mathbf{0}$ and $y^{*}>\mathbf{0}$ need not be fulfilled.
If the general eigenspace associated to $r$ is not one-dimensional the
behaviour of the semigroup $\\{T(t)\\}_{t\geq 0}$ is more complex. For
example, if $Q$ is the $2\times 2$ matrix with $q_{11}=q_{12}=q_{22}=1$ and
$q_{21}=0$, then $r=1$ is the dominant eigenvalue and
$T(t)x=e^{t}(x_{1},x_{2}+x_{1}t)$. This means that
$\lim_{t\to\infty}e^{-t}t^{-1}T(t)x=xR$, where $xR=(0,x_{1})$. Generally, if
(P) holds, then the semigroup $\\{T(t)\\}_{t\geq 0}$ has polynomial
exponential growth, i.e. there exists a nonnegative integer $k$ and a $n\times
n$ nonzero matrix $R$ with nonnegative elements such that, for each $x$,
$\lim_{t\to\infty}e^{-r\,t}t^{1-k}T(t)x=xR$. More precisely, $k\times k$ is
the size of the largest Jordan block related to the eigenvalue $r$, and $xR$
is a linear combination of certain eigenvectors related to these Jordan
blocks.

If (P) is not satisfied, a dominant eigenvalue, defined as an eigenvalue with
the largest real part, may be complex, and there may exist a number of
eigenvalues with this property. These two new features result in two new types
of asymptotic behaviour: using the Jordan–Chevalley decomposition of $Q$ we
show that (a) if imaginary parts of all dominant eigenvalues are
commensurable, then the semigroup has a periodic polynomial exponential growth
and (b) if they are not, the semigroup has an almost periodic exponential
growth. To summarise: asymptotic behaviour of the semigroup $\\{T(t)\\}_{t\geq
0}$ is determined by its behaviour on the invariant subspace $E_{d}$, spanned
by generalised eigenspaces, with the largest dimension, related to dominant
eigenvalues.

If we turn to semigroups in infinite-dimensional spaces, new problems appear
because spectral properties of infinite-dimensional semigroups are more
complex. For example, consider the intensity matrix $Q$ indexed by integers,
with elements $q_{i\,i+1}=q_{i\,i-1}=1$, $q_{ii}=-2$ and $q_{ij}=0$ in other
cases. Since this $Q$ is related to a bounded linear operator on the space
$l^{1}$, we may think of the semigroup $\\{T(t)\\}_{t\geq 0}$ defined by
$T(t)=e^{tQ}$. It can be shown that $\\{T(t)\\}_{t\geq 0}$ is a stochastic
semigroup, i.e. $\|T(t)x\|=\|x\|$ for $x\geq 0$. The matrix $Q$ also satisfies
conditions (P) and (I) and thus one could expect that there exists
$x^{*}>\mathbf{0}$ such that $T(t)x^{*}=x^{*}$, but that is not the case. In
other words, for any $t$, $r=1$ is not an eigenvalue of $T(t)$, even though
the spectral radius of $T(t)$ is $1$, and thus condition (40) fails to hold.
The main reasons for this is non-compactness of trajectories $t\mapsto T(t)x$.

The second problem in infinite-dimensional spaces is related to the rate of
convergence in (40). Let $\\{T(t)\\}_{t\geq 0}$ be a semigroup on a Banach
space $E$. We say that this semigroup has asynchronous (or balanced)
exponential growth if there exist $r\in\mathbb{C}$, nonzero $x^{*}\in E$, and
a nonzero linear functional $\alpha\colon E\to\mathbb{C}$ such that

$\lim_{t\to\infty}e^{-r\,t}T(t)x=x^{*}\alpha(x)\quad\textrm{for $x\in E$}.$
(41)

This definition implies that $r$ is a simple eigenvalue of the infinitesimal
generator $A$ of the semigroup $\\{T(t)\\}_{t\geq 0}$. In finite-dimensional
cases, convergence in (41) is exponential because the spectrum of $A$ is
finite and all eigenvalues $\lambda$ of $A$, other than $r$, satisfy the
inequality $\mathrm{Re\,}\lambda<\mathrm{Re\,}r$. But in the infinite-
dimensional case, condition (41) does not guarantee that $r$ is an isolated
element of the spectrum of $A$ and the rate of convergence in (41) needs not
be exponential.

Finally, note that if $E_{0}=\mathrm{Ker\,}\alpha$, then $E_{0}$ is invariant
with respect to the semigroup. By restricting $\\{T(t)\\}_{t\geq 0}$ to this
subspace, the problem of the rate of convergence (41) is replaced by that of
the rate of convergence of $\|e^{-r\,t}T(t)x\|\to 0\,$ for $x\in E_{0}$.

###  4.2 Asymptotic decomposition via spectral properties

There are several results on asymptotic behaviour of strongly continuous
semigroups which are in the spirit of the previous section. These results are
based on compactness properties of semigroups and are thoroughly discussed in
monographs EngelNagel ; Nagel .

Let $A$ be the infinitesimal generator of a strongly continuous semigroup
$\\{T(t)\\}_{t\geq 0}$ on a Banach space $E$ and let $\mu\in\mathbb{C}$ be an
isolated point of the spectrum of $A$. Since the resolvent $R_{\lambda}$ is a
holomorphic function of $\lambda$, it can be expanded as a Laurent series

$R_{\lambda}=\sum_{n=-\infty}^{\infty}(\lambda-\mu)^{n}U_{n}$

for $0<|\lambda-\mu|<\delta$ and some sufficiently small $\delta>0$. The
coefficients $U_{n}$ of this series are bounded operators given by the
formulas

$U_{n}=\frac{1}{2\pi
i}\oint_{\gamma}\frac{R_{\lambda}}{(\lambda-\mu)^{n+1}}\,d\lambda,\quad
n\in\mathbb{Z},$

where $\gamma$ is the positively oriented boundary of a disc with radius
$<\delta$ and center at $\mu$. The operator $P=U_{-1}$ is called the residue
of $R_{\lambda}$ at $\mu$. We have $U_{-(n+1)}=(A-\mu I)^{n}P$. If there
exists a $k>0$ such that $U_{-k}\neq 0$ while $U_{-n}=0$ for all $n>k$, then
$\mu$ is called a pole of $R_{\lambda}$ of order $k$.

Let $(L(E),\|\cdot\|)$ be the space of all bounded linear operators on $E$
endowed with the operator norm. Denote by $K(E)\subseteq L(E)$ the set of all
compact operators on $E$. A strongly continuous semigroup $\\{T(t)\\}_{t\geq
0}$ on $E$ is called eventually compact if there exists $t_{0}>0$ such that
the operator $T(t_{0})$ is compact. It can be shown that an eventually compact
semigroup is norm continuous for $t\geq t_{0}$. A strongly continuous
semigroup $\\{T(t)\\}_{t\geq 0}$ on $E$ is called quasi-compact if

$\lim_{t\to\infty}\inf\\{\|T(t)-K\|\colon K\in K(E)\\}=0.$

It is clear that an eventually compact semigroup is quasi-compact but not vice
versa. The asymptotic behaviour of a quasi-compact semigroup is fully
characterized by the following theorem (see (EngelNagel, , p. 333)).

###### Theorem 4.1

Let $\\{T(t)\\}_{t\geq 0}$ be a quasi-compact semigroup on a Banach space $E$
with generator $A$. Then the set
$\\{\lambda\in\sigma(A)\colon\mathrm{Re\,}\lambda\geq 0\\}$ is finite (or
empty) and consists of poles of $R_{\lambda}$ of finite algebraic
multiplicity. If we denote these poles by $\lambda_{1},\dots,\lambda_{m}$, and
their residues by $P_{1},\dots,P_{m}$ with poles of order $k_{1},\dots,k_{m}$,
we have $T(t)=T_{1}(t)+T_{2}(t)+\dots+T_{m}(t)+R(t)$, where

$T_{n}(t)=e^{\lambda_{n}t}\sum_{j=0}^{k_{n}-1}\frac{t^{j}}{j!}(A-\lambda_{n}I)^{j}P_{n},\quad\textrm{for
$t\geq 0$ and $1\leq n\leq m$},$ (42)

and

$\|R(t)\|\leq Me^{-\varepsilon t}\quad\textrm{for some $\varepsilon>0$, $M\geq
1$ and all $t\geq 0$}.$ (43)

For a particular instance of this theorem, consider the case when there exists
a unique dominant eigenvalue $r\in\mathbb{C}$ being a first-order pole. Then
$r$ is the spectral bound $s(A)$ of $A$, i.e.

$s(A)=\sup\\{\textrm{Re\,}\lambda\colon\lambda\in\sigma(A)\\}$

and $P=\lim\limits_{\lambda\to r}(\lambda-r)R_{\lambda}$. From Theorem 4.1 it
follows that

$\|e^{-r\,t}T(t)-P\|\leq Me^{-\varepsilon t}\quad\textrm{for some
$\varepsilon>0$, $M\geq 1$ and all $t\geq 0$}.$ (44)

Theorem 4.1 is also a proper background for a generalisation of the Perron-
Frobenius theorem to positive semigroups on Banach lattices. To obtain a
version of the Perron–Frobenius theorem for such a semigroup, one typically
checks, instead of (I), that the semigroup is _irreducible_ ; this means that
for every $x\in E_{+}\setminus\\{\mathbf{0}\\}$ and for every $x^{*}\in
E^{*}_{+}\setminus\\{\mathbf{0}\\}$ there exists a $t>0$ such that
$x^{*}(T(t)x)>0$ (here $E_{+}=\\{x\in E\colon x\geq 0\\}$ and
$E^{*}_{+}=\\{x\in E^{*}\colon x\geq 0\\}$). Indeed, the irreducibility rules
out the possibility for the range of $P$ to be more than one-dimensional, and
together with quasi-compactness implies asynchronous exponential growth of the
semigroup. A similar result can be obtained if we replace quasi-compactness by
an assumption based on the notion of the measure of noncompactness of $A$, see
e.g. Webb-a .

The asynchronous exponential growth (41) is a special case of the situation
when there exists the limit

$\lim_{t\to\infty}e^{-r\,t}T(t)=P,$ (45)

where $P\colon E\to E$ is an arbitrary nonzero operator, and the convergence
can be uniform, strong or weak. The latter property was studied in Thieme-a ;
Thieme-b , and a comprehensive overview of applications of this theory to
population dynamics can be found in Webb-b . If
$\widetilde{T}(t)=e^{-r\,t}T(t)$ and $\widetilde{E}=\mathrm{Ker}\,P$, then
condition (45) in the case of strong convergence is equivalent to strong
asymptotic stability of the semigroup $\\{\widetilde{T}(t)\\}_{t\geq 0}$, i.e.
to the condition

$\lim_{t\to\infty}\|\widetilde{T}(t)x\|=0\quad\textrm{for
$x\in\widetilde{E}$}.$ (46)

In the case when the semigroup is bounded and the set
$\sigma(A)\cap(i\mathbb{R})$ is at most countable, the strong asymptotic
stability holds if and only if the adjoint operator $A^{*}$ has no pure
imaginary eigenvalues ArendtBatty ; Lyubich-Phong . Some versions of this
result for unbounded operators are proved in SklyarPolak .

The question of deriving the optimal rate of decay in (46) is highly
nontrivial. This problem is extensively discussed in BattyChillTomilov . To
give a taste of this theory we recall one of the results of BorichevTomilov
(Theorem 2.4). Let $\\{T(t)\\}_{t\geq 0}$ be a strongly continuous semigroup
on a Hilbert space $E$ with generator A such that
$\sigma(A)\cap(i\mathbb{R})=\emptyset$. Then for a fixed $\alpha>0$ the
following conditions are equivalent:

  1. (i)

$R_{is}=O(|s|^{\alpha})$, $s\to\infty$,

  2. (ii)

$\|T(t)A^{-1}\|=O(t^{-\alpha})$, $t\to\infty$,

  3. (iii)

$\|T(t)A^{-1}x\|=o(t^{-\alpha})$, $t\to\infty$, $x\in E$.

###  4.3 Asymptotics of stochastic semigroups: introductory results

We will consider stochastic semigroups on the space $L^{1}=L^{1}(X,\Sigma,m)$
with a $\sigma$-finite measure $m$. The iterates of a stochastic operator form
a _discrete-time_ semigroup, and by writing $P(t)=P^{t}$ for these iterates,
we may formulate most of definitions and results for both types of semigroups
in a unified manner.

We will begin by studying the asymptotics of uniformly continuous stochastic
semigroups. We recall that such a semigroup is given by the formula

$P(t)f=\sum_{k=0}^{\infty}\frac{(\lambda t)^{k}e^{-\lambda t}}{k!}P^{k}f,$
(47)

where $P$ is a stochastic operator and $\lambda\geq 0$.

###### Theorem 4.2

If $Pf^{*}=f^{*}$ for some density $f^{*}>0$ a.e., then for any density $f$
there exists an invariant density $\bar{f}$ such that
$\bar{f}=\lim_{t\to\infty}P(t)f$ in $L^{1}$. If additionally, $f^{*}$ is a
unique invariant density of $P$, then $\lim_{t\to\infty}P(t)f=f^{*}$.

The proof of this theorem is given in PR-prz-apss-apdMp . In the proof we
define a new operator $\tilde{P}f=P(f\\!f^{*})/f^{*}$ on the probability space
$L^{1}(X,\Sigma,\mu)$ with $\mu(dx)=f^{*}(x)\,m(dx)$. Then $\tilde{P}$ is a
double stochastic operator, i.e. $\tilde{P}$ is a stochastic operator and
$\tilde{P}\mathbf{1}_{X}=\mathbf{1}_{X}$. Next we show that Theorem 4.2
results from the following lemma

###### Lemma 1

If $P$ is a double stochastic operator, then
$P\operatorname{E}(f|\Sigma_{\mathrm{inv}})=\operatorname{E}(f|\Sigma_{\mathrm{inv}})$
and

$\lim_{t\to\infty}\|e^{t(P-I)}f-\operatorname{E}(f|\Sigma_{\mathrm{inv}})\|=0\quad\textrm{for
$f\in L^{1}(X,\Sigma,\mu)$.}$

The $\sigma$-algebra $\Sigma_{\mathrm{inv}}$ is given by
$\Sigma_{\mathrm{inv}}=\\{A\in\Sigma\colon
P^{*}\mathbf{1}_{A}=\mathbf{1}_{A}\\}$.

A stochastic semigroup $\\{P(t)\\}_{t\geq 0}$ is called asymptotically stable
if there exists a density $f^{*}$ such that

$\lim_{t\to\infty}\|P(t)f-f^{*}\|=0\quad\textrm{for}\quad f\in D.$ (48)

We recall that $D$ denotes the set of all densities. From (48) it follows that
$f^{*}$ is an invariant density, i.e. $P(t)f^{*}=f^{*}$ for $t\geq 0$. If the
semigroup $\\{P(t)\\}_{t\geq 0}$ is generated by an evolution equation
$u^{\prime}(t)=Au(t)$, then its asymptotic stability means that the stationary
solution $u(t)=f^{*}$ is globally asymptotically stable in the sense of
Lyapunov on the set of densities $D$. Observe that if there is an invariant
density $f^{*}$, then the stochastic semigroup $\\{P(t)\\}_{t\geq 0}$ is
asymptotically stable if and only if $\lim_{t\to\infty}\|P(t)f\|=0$ for $f\in
L^{1}_{0}=\\{f\in L^{1}\colon\int_{X}f(x)\,m(dx)=0\\}$.

There are several results concerning asymptotic stability of stochastic
semigroups. We start with the lower function theorem of Lasota and Yorke LY .
Let $f^{-}(x)=\max(0,-f(x))$. A function $h\in L^{1}$, such that $h\geq 0$ and
$h\neq 0$, is called a lower function for a stochastic semigroup
$\\{P(t)\\}_{t\geq 0}$ if

$\lim_{t\to\infty}\|(P(t)f-h)^{-}\|=0\quad\textrm{for every $f\in D$.}$ (49)

###### Theorem 4.3

A continuous or discrete-time stochastic semigroup is asymptotically stable if
and only if it has a lower function $h$.

This theorem was generalized in many directions. The interested reader is
referred to the book Emelyanov , where one can find versions formulated in the
language of von Neumann algebras and to the paper Gerlach-Gluck-ETDS  which
provides a comprehensive overview of earlier results and presents new theorems
for semigroups on Banach lattices including a version with lower functions
depending on $f$.

###  4.4 Partially integral semigroups

In next two sections we present two theorems on asymptotic behaviour of
stochastic semigroups, that stem from the theory of Harris operators Fo ;
Meyn-Tweedie ; Num .

A stochastic semigroup $\\{P(t)\\}_{t\geq 0}$ is called partially integral if
there exist a $t>0$ and a measurable function $q(t,\cdot,\cdot)\colon X\times
X\to[0,\infty)$, called the kernel of the semigroup, such that

$\int_{X}\int_{X}q(t,x,y)\,m(dx)\,m(dy)>0$

and

$P(t)f(y)\geq\int_{X}q(t,x,y)f(x)\,m(dx)\quad\textrm{for $f\in D$.}$ (50)

###### Theorem 4.4

Let $\\{P(t)\\}_{t\geq 0}$ be a partially integral stochastic semigroup. We
assume that this semigroup has invariant density $f^{*}$ and there are no
other periodic points in the set of densities. If $f_{*}>0$ a.e., then the
semigroup $\\{P(t)\\}_{t\geq 0}$ is asymptotically stable.

The proof of Theorem 4.4 is given in R-b95 . If $\\{P(t)\\}_{t\geq 0}$ is a
continuous-time stochastic semigroup, then Theorem 4.4 can be strengthened
significantly PR-jmaa2 :

###### Theorem 4.5

Suppose that a continuous-time partially integral stochastic semigroup
$\\{P(t)\\}_{t\geq 0}$ has a unique invariant density $f^{*}$. If $f^{*}>0$
a.e., then the semigroup $\\{P(t)\\}_{t\geq 0}$ is asymptotically stable.

###### Corollary 1

If a continuous-time partially integral stochastic semigroup is irreducible
and has an invariant density, then it is asymptotically stable.

In order to formulate other corollaries to Theorem 4.4, we need two auxiliary
definitions.

We say that a stochastic semigroup $\\{P(t)\\}_{t\geq 0}$ spreads supports  if
for every set $A\in\Sigma$ and for every $f\in D$ we have

$\lim_{t\to\infty}m(\mathrm{supp}\,P(t)f\cap A)=m(A)$

and overlaps supports, if for every $f,g\in D\,$ there exists $t>0$ such that

$m(\mathrm{supp}\,P(t)f\cap\mathrm{supp}\,P(t)g)>0.$

###### Corollary 2

A partially integral stochastic semigroup which spreads supports and has an
invariant density is asymptotically stable.

###### Corollary 3

A partially integral stochastic semigroup which overlaps supports and has an
invariant density $f_{*}>0$ a.e. is asymptotically stable.

We will now introduce the notion of sweeping, which, together with asymptotic
stability, will allow us to give a characterisation of the behaviour of a
rather wide class of substochastic semigroups.

A stochastic semigroup $\\{P(t)\\}_{t\geq 0}$ is called sweeping with respect
to a set $B\in\Sigma$ if for every $f\in D$

$\lim_{t\to\infty}\int_{B}P(t)f(x)\,m(dx)=0.$

The notion of sweeping was introduced in KT  but it is also known as zero type
property. In KT  the following theorem was proved.

###### Theorem 4.6

Let $\\{P(t)\\}_{t\geq 0}$ be an integral stochastic semigroup. Assume that
this semigroup has no invariant density, while for a certain set $A\in\Sigma$
the following condition holds: (KT) there exists a measurable function $f^{*}$
such that $0<f^{*}<\infty$ a.e. and $P(t)f^{*}\leq f^{*}$ for $t\geq 0$,
$f^{*}\notin L^{1}$ and $\,\int_{A}f^{*}(x)\,m(dx)<\infty$. Then the semigroup
$\\{P(t)\\}_{t\geq 0}$ is sweeping with respect to $A$.

In paper R-b95  it was shown that Theorem 4.6 holds for a broader class of
semigroups than integrals, for example to overlapping semigroups.

###### Theorem 4.7

Let $\\{P(t)\\}_{t\geq 0}$ be a stochastic semigroup which overlaps supports.
Assume that the semigroup $\\{P(t)\\}_{t\geq 0}$ and a set $A\in\Sigma$
satisfy condition (KT). Then the semigroup $\\{P(t)\\}_{t\geq 0}$ is sweeping
with respect to $A$.

The advantage of the theorems 4.6 and 4.7 is that knowing the function $f^{*}$
we can determine the sets from which the semigroup is sweeping. The main
problem in applying the theorems 4.6 and 4.7 is to prove that a stochastic
semigroup satisfies condition (KT).

###  4.5 Asymptotic decomposition via kernel minorants

Theorems 4.4 and 4.5 allow us to prove the asymptotic stability of Markov
semigroups when we know that there is a positive invariant density.
Unfortunately, in most applications it is difficult to determine the invariant
density or even to prove its existence.

Our immediate goal will be to present results on the asymptotic behaviour of
substochastic semigroups, in which we do not directly assume that the
semigroup has invariant density. One of the corollaries will be a criterion of
sweeping from compact sets, not based on condition (KT) and convenient for
applications.

We present a method developed in PR-JMMA2016 ; PR-SD  to study asymptotic
properties of stochastic semigroups via their asymptotic decomposition. From
now on we assume additionally that $(X,\rho)$ is a separable metric space and
$\Sigma=\mathcal{B}(X)$ is the $\sigma$-algebra of Borel subsets of $X$, and
consider a partially integral substochastic semigroups $\\{P(t)\\}_{t\geq 0}$
with a kernel $q$ which satisfies the following condition:

  1. (K)

for every $x_{0}\in X$ there exist an $\varepsilon>0$, a $t>0$, and a
measurable function $\eta\geq 0$ such that $\int\eta(x)\,m(dx)>0$ and

$q(t,x,y)\geq\eta(y)\quad\textrm{for $x\in B(x_{0},\varepsilon)$, $y\in X$},$
(51)

where $B(x_{0},\varepsilon)=\\{x\in X:\,\,\rho(x,x_{0})<\varepsilon\\}$.

It is clear that substochastic semigroups satisfying condition (K) are
partially integral.

Now we present two theorems on asymptotic decomposition: the first for a
continuous-time substochastic semigroups and the second for substochastic
operators (discrete-time semigroups) PR-JMMA2016 ; PR-SD .

###### Theorem 4.8

Let $\\{P(t)\\}_{t\geq 0}$ be a continuous-time substochastic semigroup which
satisfies (K). Then there exist an at most countable set $J$, a family of
invariant densities $\\{f^{*}_{j}\\}_{j\in J}$ with disjoint supports
$\\{A_{j}\\}_{j\in J}$, and a family $\\{\alpha_{j}\\}_{j\in J}$ of positive
linear functionals defined on $L^{1}$ such that

  1. (i)

for every $j\in J$ and for every $f\in L^{1}$ we have

$\lim_{t\to\infty}\|\mathbf{1}_{A_{j}}P(t)f-\alpha_{j}(f)f^{*}_{j}\|=0,$ (52)

  2. (ii)

if $\,Y=X\setminus\bigcup\limits_{j\in J}A_{j}$, then for every $f\in L^{1}$
and for every compact set $F$ we have

$\lim_{t\to\infty}\int\limits_{F\cap Y}P(t)f(x)\,m(dx)=0.$ (53)

We note that not only the sets $A_{j}$, $j\in J$, are disjoint but also their
closures are disjoint, which means that the measures
$\mu_{j}(dx)=f^{*}_{j}(x)\,m(dx)$ have disjoint topological supports PR-SD .

We will precede the formulation of the asymptotic decomposition for
substochastic operators with some auxiliary definitions.

Let $E$ be a vector space. A linear mapping $T\colon E\to E$ satisfying the
condition $T^{2}=T$ is called a  projection. If $y\in T(E)$, then $y=Tx$ for
some $x\in E$ and from the definition of projection we get $Ty=T^{2}x=Tx=y$,
so $T$ is the identity on $T(E)$.

Let $(X,\Sigma,m)$ be a $\sigma$-finite measure space. A substochastic
operator $S$ is called periodic on the set $B\in\Sigma$, if there is a
sequence of densities $h_{1},\dots,h_{k}$ satisfying the following conditions

$\displaystyle h_{i}(x)=0\quad\text{for $i=1,\dots,k$ and $x\notin B$},$ (54)
$\displaystyle h_{i}h_{j}=0\quad\text{for $i\neq j$,\quad\
$h_{1}(x)+\cdots+h_{k}(x)>0$\quad for $x\in B$,}$ (55) $\displaystyle
Sh_{i}=h_{i+1}\quad\text{for $i\leq k$, \quad where\quad$h_{k+1}:=h_{1}$,}$
(56)

and $Sf=ST\\!f$ for $f\in L^{1}(X,\Sigma,m)$, where $T$ is a positive
projection from the space $L^{1}(X,\Sigma,m)$ onto its linear subspace spanned
by functions $h_{1},\dots,h_{k}$.

###### Theorem 4.9

Suppose that the substochastic operator $P$ has the property (K). Then there
exist at most countable set $J$, a family $\\{A_{j}\\}_{j\in J}$ of pairwise
disjoint measurable subsets of $X$ and the family of substochastic operators
$\\{S_{j}\\}_{j\in J}$, satisfying the conditions:

  1. (i)

$P^{*}\mathbf{1}_{A_{j}}\geq\mathbf{1}_{A_{j}}$ for $j\in J$;

  2. (ii)

for each $j\in J$ the operator $S_{j}$ is periodic on the set $A_{j}$;

  3. (iii)

for each $j\in J$ and for any function $f\in L^{1}(X,\Sigma,m)$ we have

$\lim_{n\to\infty}\|\mathbf{1}_{A_{j}}P^{n}\\!f-S^{n}_{j}f\|=0;$ (57)

  4. (iv)

if $Y=X\setminus\bigcup_{j\in J}A_{j}$, then for any function $f\in
L^{1}(X,\Sigma,m)$ and any compact set $F\subset X$ we have

$\lim_{n\to\infty}\int\limits_{F\cap Y}P^{n}\\!f(x)\,m(dx)=0.$ (58)

###### Remark 4

From (i) it follows that the operator $P$ can be restricted to the space
$L^{1}(A_{j})$ and $P$ is a stochastic operator on this space. Also the
operator $S_{j}$ restricted to the space $L^{1}(A_{j})$ is a stochastic
operator and this restriction is given by the formula

$S_{j}f=\sum_{i=1}^{k}\alpha_{i}(f)h_{i+1},\quad\text{$\alpha_{i}(f)=\int_{B_{i}}f(x)\,m(dx)$,
$\,B_{i}=\operatorname{supp}h_{i}$,}$ (59)

where the functions $h_{1},\dots,h_{k}$ are the densities appearing in the
definition of periodic operator. The operator $P$ satisfies (56), i.e.
$Ph_{i}=h_{i+1}$.

The following criterion of sweeping follows from Theorems 4.8 and 4.9.

###### Corollary 4

Assume that a substochastic (continuous or discrete-time) semigroup
$\\{P(t)\\}_{t\geq 0}$ satisfies condition (K) and has no invariant densities.
Then $\\{P(t)\\}_{t\geq 0}$ is sweeping from compact sets.

Our next aim is to find conditions guaranteeing that a stochastic semigroup
$\\{P(t)\\}_{t\geq 0}$ satisfies the Foguel alternative, i.e. it is
asymptotically stable or sweeping from all compact sets LiM . The following
result is a step in this direction.

###### Corollary 5

If a continuous-time substochastic semigroup $\\{P(t)\\}_{t\geq 0}$ is
irreducible and satisfies condition (K), then the Foguel alternative holds.

In the case of substochastic operator we have a slightly weaker version of
this corollary. A substochastic operator $P$ is called asymptotically periodic
if there exists a periodic operator $S$ such that

$\lim_{n\to\infty}\|P^{n}f-S^{n}f\|=0\quad\text{for $f\in D$}.$

###### Corollary 6

If a substochastic operator satisfies (K) and is irreducible, then it is
either asymptotically periodic or sweeping from compact sets.

Now we apply Corollary 5 to Markov chains. Let $Q$ be the intensity matrix of
a continuous-time Markov chain $(\xi_{t})_{t\geq 0}$ on $\mathbb{N}$. Since
the space $\mathbb{N}$ is discrete the stochastic semigroup $\\{P(t)\\}_{t\geq
0}$ related to the process $(\xi_{t})_{t\geq 0}$ satisfies condition (K).

###### Theorem 4.10

Let $\\{P(t)\\}_{t\geq 0}$ be the stochastic semigroup on $l^{1}$ related to a
continuous-time Markov chain $(\xi_{t})_{t\geq 0}$ with intensity matrix $Q$.
Let us assume that the entries of the matrix $Q$ satisfy the following
condition

  1. (I)

for all $i$ and $j$ there exists a sequence $(i_{1},i_{2},\dots,i_{m})$ such
that $i_{1}=i$, $i_{m}=j$ and $q_{i_{k}i_{k+1}}>0\,$ for $k=1,\dots,m-1$.

Then the semigroup $\\{P(t)\\}_{t\geq 0}$ satisfies the Foguel alternative:

  1. (a)

if the semigroup $\\{P(t)\\}_{t\geq 0}$ has an invariant density, then it is
asymptotically stable,

  2. (b)

if the semigroup $\\{P(t)\\}_{t\geq 0}$ has no invariant density, then for
every $x\in l^{1}$ and $i\in\mathbb{N}$ we have

$\lim_{t\to\infty}(P(t)x)_{i}=0.$ (60)

If a stochastic semigroup (continuous or discrete-time) overlaps supports,
then the statements of Theorems 4.8 and 4.9 can be significantly strengthened
(see PR-cell-cycle  Proposition 1).

###### Corollary 7

Assume that a stochastic semigroup $\\{P(t)\\}_{t\geq 0}$ satisfies (K) and
overlaps supports. Then this semigroup is sweeping or has an invariant density
$f^{*}$ with a support $A$ and there exists a positive linear functional
$\alpha$ defined on $L^{1}(X,\Sigma,m)$ such that

  * (i)

for every $f\in L^{1}(X,\Sigma,m)$ we have

$\lim_{t\to\infty}\|\mathbf{1}_{A}P(t)f-\alpha(f)f^{*}\|=0,$ (61)

  * (ii)

if $\,Y=X\setminus A$, then for every $f\in L^{1}(X,\Sigma,m)$ and for every
compact set $F$ we have

$\lim_{t\to\infty}\int\limits_{F\cap Y}P(t)f(x)\,m(dx)=0.$ (62)

In particular, if $\\{P(t)\\}_{t\geq 0}$ has an invariant density $f^{*}$ with
the support $A$ and $X\setminus A$ is a subset of a compact set, then this
semigroup is asymptotically stable.

In order to formulate a further corollary, we need to introduce two additional
conditions, which are weak versions of well-known concepts: irreducibility and
tightness of measures.

  1. (WI)

there exists a point $x_{0}\in X$ such that for each $\varepsilon>0$ and for
each density $f$ we have

$\int\limits_{B(x_{0},\varepsilon)}P(t)f(x)\,m(dx)>0\quad\textrm{for some
$t>0$}.$ (63)

  1. (WT)

There exists $\kappa>0$ such that

$\sup\limits_{F\in\mathcal{F}}\limsup_{t\to\infty}\int_{F}P(t)f(x)\,m(dx)\geq\kappa$
(64)

for $f\in D_{0}$, where $D_{0}$ is a dense subset of $D$ and $\mathcal{F}$ is
the family of all compact subsets of $X$.

###### Corollary 8

Let $\\{P(t)\\}_{t\geq 0}$ be a continuous-time stochastic semigroup. Assume
that $\\{P(t)\\}_{t\geq 0}$ satisfies conditions (K), (WI), and (WT). Then the
semigroup $\\{P(t)\\}_{t\geq 0}$ is asymptotically stable.

###### Corollary 9

Suppose that $X$ is a compact space. Then a continuous-time stochastic
semigroup which satisfies (K) and (WI) is asymptotically stable.

The proof of Corollary 8 can be found in PR-SD . The second corollary follows
directly from the first one. New results concerning positive operators on
Banach lattices similar in spirit to Theorems 4.5 and 4.8 may be found in
Gerlach-Gluck .

An advantage of the formulation of some results here in the form of the Foguel
alternative is that in order to show asymptotic stability we do not need to
prove the existence of an invariant density. It is enough to check that the
semigroup is not sweeping with respect to compact sets then, automatically,
the semigroup $\\{P(t)\\}_{t\geq 0}$ is asymptotically stable. We can
eliminate the sweeping by means of some method similar to that of Lyapunov
function called Hasminskiĭ function.

###### Theorem 4.11

Let $A$ be the generator of a stochastic semigroup $\\{P(t)\\}_{t\geq 0}$,
$V\colon X\to[0,\infty)$ be a measurable function, $Z$ be a measurable set,
and $\varepsilon>0$. Assume that there exists a measurable function, formally
denoted by $LV$, satisfying conditions

$\int_{X}LV(x)g(x)\,m(dx)=\int_{X}V(x)Ag(x)\,m(dx)\quad\text{for
$g\in\mathfrak{D}(A)\cap D_{V}$},$ (65) $LV(x)\leq M-\varepsilon\quad\text{for
$\,x\in Z$},\quad\quad LV(x)\leq-\varepsilon\quad\text{for $\,x\in X\setminus
Z$},$ (66)

where $D_{V}=\\{f\in D\colon\,\int_{X}f(x)V(x)\,m(dx)<\infty\\}$. Then the
semigroup $\\{P(t)\\}_{t\geq 0}$ is not sweeping with respect to the set $Z$.
If $Z$ is compact, then the semigroup satisfies condition $(\rm{WT})$.

The main difficulty in applying this method lies in correctly defining the
action of the operator $L$ on the function $V$, since $V$ does not usually
belong to the domain of the operator $L$. The Hasminskiĭ function method has
been used to study Markov chains PRT2012 , multistate diffusion processes PR-
jmaa1 , diffusion with jumps PR-bpan , dynamical systems with jumps Pichor1998
.

###  4.6 Constrictive operators

The next theorem we discuss here shows that convergence of trajectories of a
stochastic semigroup to a compact set leads to interesting asymptotic
properties. A stochastic semigroup $\\{P(t)\\}_{t\geq 0}$ is called strongly
(weakly) constrictive LLY  if there exists a strongly (weakly) compact subset
$F$ of $L^{1}$ such that

$\lim_{t\to\infty}\inf\\{\|P(t)f-g\|\colon g\in F\\}=0\quad\text{for $f\in
D$}.$

Even though the definition of the constrictive operator and the property (K)
are significantly differ, similar conclusions follow. One of them is the
following theorem.

###### Theorem 4.12

Suppose that a stochastic operator $P$ defined on the space
$L^{1}(X,\Sigma,m)$ is constrictive. Then there exist a finite set $J$, a
family $\\{A_{j}\\}_{j\in J}$ of pairwise disjoint measurable subsets of $X$
and the family $\\{S_{j}\\}_{j\in J}$ of substochastic operators satisfying
the conditions:

  1. (i)

$P^{*}\mathbf{1}_{A_{j}}\geq\mathbf{1}_{A_{j}}$ for $j\in J$;

  2. (ii)

for each $j\in J$ the operator $S_{j}$ is periodic on the set $A_{j}$;

  3. (iii)

for each $j\in J$ and for any function $f\in L^{1}(X,\Sigma,m)$ we have

$\lim_{n\to\infty}\|\mathbf{1}_{A_{j}}P^{n}\\!f-S^{n}_{j}f\|=0;$ (67)

  4. (iv)

if $Y=X\setminus\bigcup\limits_{j\in J}A_{j}$, then for any function $f\in
L^{1}(X,\Sigma,m)$ we have

$\lim_{n\to\infty}\int\limits_{Y}P^{n}\\!f(x)\,m(dx)=0.$ (68)

Theorem 4.12 was first proved in LLY  under assumption of strong
constrictivity; two years later Komorník Komornik  showed that weak
constrictivity suffices. The original formulation of Theorem 4.12 was
different from that cited here. We have given its statement formulated in the
spirit of Theorems 4.8 and 4.9.

If there exists a function $h\in L^{1}$ such that

$\lim_{t\to\infty}\|(P(t)f-h)^{+}\|=0\quad\textrm{for every $f\in D$},$ (69)

then Theorem 4.12 holds. It is a simple consequence of the fact that the set
$F=\\{f\in D\colon f\leq h\\}$ is weakly compact. Theorem 4.12 can be
strengthened considerably in the case when $\\{P(t)\\}_{t\geq 0}$ is a
continuous-time stochastic semigroup. Then we replace condition (68) by
condition (52).

The property of constrictivness was generalised and studied in many papers
(see Emelyanov  for a comprehensive overview of these results). Examples of
applications of Theorem 4.12 and the lower function theorem are given in LiM .

##  5 Applications to biological models

###  5.1 Model of DNA evolution

The degree of relatedness between humans, species, genes and proteins, can be
studied by comparing DNA chain sequences. DNA is inherited from parents, but
mutations occur as a result of errors in DNA replication and repair. Mutations
occur throughout the DNA chain at a constant level on average, and although
the potential mutations may be numerous, as a result of natural selection,
most of them are not conserved and mainly selection-neutral mutations are
observed.

The simplest way to measure evolutionary distance is to check in how many
places the aligned DNA sequences differ. More precisely, we first compare and
align DNA sequences so that they match, and then check the differences between
them. The initial issue here is modelling nucleotide substitutions at a fixed
point in the DNA chain. Since we have four different nucleotides corresponding
to nucleobases A, G, C, T, so models describing nucleotide evolution, called
models of substitution of nucleotides in the DNA sequences are Markov chains
on a four-state space $S=\\{{\rm A,G,C,T}\\}$. The intensity matrix in this
chain is of the form

$Q=\left[\begin{matrix}q_{\rm AA}&q_{\rm AG}&q_{\rm AC}&q_{\rm AT}\\\ q_{\rm
GA}&q_{\rm GG}&q_{\rm GC}&q_{\rm GT}\\\ q_{\rm CA}&q_{\rm CG}&q_{\rm
CC}&q_{\rm CT}\\\ q_{\rm TA}&q_{\rm TG}&q_{\rm TC}&q_{\rm
TT}\end{matrix}\right].$

The simplest substitution model is Jukes–Cantor model JC . In this model, we
assume that probability of replacing any nucleotide, by another nucleotide is
the same. The intensity matrix $Q$ in this model is of the form

$Q=\left[\begin{matrix}-3\lambda&\lambda&\lambda&\lambda\\\
\lambda&-3\lambda&\lambda&\lambda\\\ \lambda&\lambda&-3\lambda&\lambda\\\
\lambda&\lambda&\lambda&-3\lambda\end{matrix}\right].$

Let $p_{ij}(t)={\rm P}(\xi_{t}=j|\xi_{0}=i)$. It is easy to check that

$p_{ij}(t)=\frac{1}{4}-\frac{1}{4}e^{-4\lambda t}\text{ \,\,for $i\neq
j$,}\quad\ p_{ii}(t)=\frac{1}{4}+\frac{3}{4}e^{-4\lambda t}\text{ \,\,for
$i\in S$.}$

These formulas allow us to estimate the evolutionary distance of two different
individuals or species based on differences in aligned DNA sequences. We will
assume that both sequences are derived from a common ancestor and that, as a
result of mutations, there have been independent changes in both sequences in
successive generations. As a measure of evolutionary distance, we will take
the time elapsed since the common ancestor, or the number of generations.

We assume that a mutation of a single nucleotide occurred on average every
$t_{0}$ years. To begin, we compare the sequence of an ancestor that lived $t$
years ago with that of a contemporary living individual. Since the intensity
of nucleotide change is $3\lambda$, so $3\lambda t_{0}=1$. The probability of
changing a nucleotide at a fixed site is

$p=3\bigl{(}\tfrac{1}{4}-\tfrac{1}{4}e^{-4\lambda t}\bigr{)}.$

Since $3\lambda t_{0}=1$, we have

$\frac{t}{t_{0}}=-\tfrac{3}{4}\ln\big{(}1-\tfrac{4}{3}p\big{)}.$ (70)

If the sequences are of length $n$ and we have $n_{d}$ differences, we assume
that $\hat{p}=n_{d}/n$ is the estimated value of $p$.

Let us now consider two sequences of contemporary individuals. Suppose we want
to determine how many years ago their common ancestor lived. The Markov chain
in this model is reversible. Thanks to this symmetry, we can treat one of the
individuals as the ,,ancestor” of their common ancestor. Thus, the time to the
common ancestor will be twice as small as the determined by the formula (70).

###  5.2 Growth of erythrocyte population

Consider the following model of erythrocyte population growth (red blood
cells). Erythrocytes are formed in the bone marrow by the division and
differentiation of erythroid stem cells. Mature erythrocytes enter the
bloodstream, where they serve mainly to transport oxygen to the cells and are
destroyed after about four months. Assume that $b>0$ is the intensity of
erythrocyte production in the bone marrow, and $d>0$ is the destruction rate
of a single erythrocyte. We check that the number of erythrocytes in the
bloodstream is described by a Markov chain (a certain birth-death process) on
the space $\mathbb{N}$. The intensity matrix $Q$ has entries: $q_{i\,i+1}=b$,
$q_{i\,i-1}=di$, and $q_{ii}=-(b+di)$ for $i\in\mathbb{N}$ and $q_{ij}=0$
otherwise.

First, using Theorem 3.4 we prove that the matrix $Q$ generates a stochastic
semigroup $\\{P(t)\\}_{t\geq 0}$. Let $\lambda>0$ be a positive constant and
assume a contrary that there exists a nonzero and nonnegative $x\in
l^{\infty}$ which is a solution of the equation $Qx=\lambda x$. Then

$dix_{i-1}-(b+di)x_{i}+bx_{i+1}=\lambda x_{i}\quad\text{for
$\,i\in\mathbb{N}$},$ (71)

where we formally assume that $x_{-1}=0$. Then

$x_{i+1}-x_{i}=\frac{\lambda}{b}x_{i}+\frac{d}{b}(x_{i}-x_{i-1})i.$ (72)

Hence, the sequence $(x_{i})$ is increasing and
$x_{i+1}\geq\big{(}1+\frac{\lambda}{b}\big{)}x_{i}$. Consequently, $x\notin
l^{\infty}$, and the matrix $Q$ generates a stochastic semigroup.

We show that the semigroup $\\{P(t)\\}$ has an invariant density. Since the
generator $A$ of this semigroup is given by the formula $Ax=xQ$, we are
looking for a positive solution of the equation $xQ=0$. Then

$bx_{i-1}-(b+di)x_{i}+d(i+1)x_{i+1}=0\quad\text{for $\,i\in\mathbb{N}$},$ (73)

where $x_{-1}=0$. The sequence $x_{i}=b^{i}/(d^{i}i!)$ satisfies (73). Since
$\sum_{i=0}^{\infty}x_{i}=e^{b/d}$, in order to obtain a probabilistic
distribution we divide $x_{i}$ by $e^{b/d}$. This will yield the invariant
density $(\pi_{i})$, which is a Poisson distribution with the parameter
$\lambda=b/d$:

$\pi_{i}=\frac{\lambda^{i}e^{-\lambda}}{i!}\quad\text{for
$\,i\in\mathbb{N}$.}$

Since the matrix $Q$ satisfies condition (I) of Theorem 4.10, the semigroup
$\\{P(t)\\}_{t\geq 0}$ is asymptotically stable.

###  5.3 Population growth with stochastic noise

Consider the following equation

$dx_{t}=b(x_{t})\,dt+\sigma(x_{t})\,dw_{t}.$ (74)

Eq. (74) is a stochastic version of the deterministic models describing the
growth of a single population $x^{\prime}(t)=b(x)$, where $x(t)$ is the size
of the population at time $t$ and the function $b$ is the growth coefficient.
We assume that $b\colon[0,\infty)\to\mathbb{R}$ is a $C^{1}$-function on the
whole interval $[0,\infty)$ satisfying additionally the conditions $b(0)=0$
and $b(x)\leq Kx$ for $x\geq 0$ and some $K>0$.

We can consider two different stochastic perturbations. The first, called
environmental noise, is used to describe small populations in which random
fluctuations act in the same way on all individuals in the population, so
$\sigma(x)=\sigma x$, $\sigma>0$. The second is called demographic noise. In
this case the stochastic perturbation acts independently on each part of
population and it has zero mean. According to the central limit theorem such
noise is proportional to $\sqrt{x}$, so $\sigma(x)=\sigma\sqrt{x}$. Models
with demographic noise are more difficult to study because the trajectories of
the process $(x_{t})$ are absorbed with positive probability at $0$.

We only consider here models with environmental noise, thus Eq. (74) has the
form

$dx_{t}=b(x_{t})\,dt+\sigma x_{t}\,dw_{t}.$ (75)

The stochastic semigroup $\\{P(t)\\}_{t\geq 0}$ related to the process
$(x_{t})_{t\geq 0}$ is integral and the kernel $q(t,x,y)$ is positive and
continuous for $t,x,y>0$. Thus, the semigroup $\\{P(t)\\}_{t\geq 0}$ is
irreducible and satisfies condition (K). Then

$\frac{\partial u}{\partial t}=\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}(\sigma^{2}x^{2}u)-\frac{\partial}{\partial x}(b(x)u)$ (76)

is the Fokker–Planck equation corresponding to this semigroup. An invariant
density $f^{*}$, if it exists, is a solution of the equation

$\frac{1}{2}\big{(}\sigma^{2}x^{2}f^{*}(x)\big{)}^{\prime\prime}-\big{(}b(x)f^{*}(x)\big{)}^{\prime}=0.$
(77)

We easily determine from (77) the invariant density

$f^{*}(x)=\frac{C}{x^{2}}\exp\bigg{(}\int_{1}^{x}\frac{2b(s)}{\sigma^{2}s^{2}}\,ds\bigg{)},$
(78)

as long as $C$ can be chosen such that $f^{*}$ is a density. Thus, an
invariant density exists, and consequently the semigroup $\\{P(t)\\}_{t\geq
0}$ is asymptotically stable (see Corollary 1), if and only if

$\int_{0}^{\infty}\frac{1}{x^{2}}\exp\bigg{(}\int_{1}^{x}\frac{2b(s)}{\sigma^{2}s^{2}}\,ds\bigg{)}\,dx<\infty.$
(79)

If (79) does not hold then according to Corollary 4 the semigroup
$\\{P(t)\\}_{t\geq 0}$ is sweeping from intervals $[\alpha,\beta]$,
$0<\alpha<\beta<\infty$, i.e.

$\lim_{t\to\infty}\int_{\alpha}^{\beta}P(t)f(x)\,\,dx=0\quad\text{for $f\in
D$.}$ (80)

Using Theorem 4.6, we can show more precise sweeping conditions and give their
biological interpretation. If (79) does not hold, then $f^{*}$ given by (78)
with $C>0$ is a non-integrable positive function such that $P(t)f^{*}=f^{*}$.
Assume that there exists the limit
$b^{\prime}(\infty)=\lim_{x\to\infty}b(x)/x$. It is easy to check that

$\displaystyle\int_{0}^{\alpha}f^{*}(x)\,dx<\infty\textrm{ if
$\sigma^{2}<2b^{\prime}(0)$},$
$\displaystyle\int_{0}^{\alpha}f^{*}(x)\,dx=\infty\textrm{ if
$\sigma^{2}>2b^{\prime}(0)$},$ (81)
$\displaystyle\int_{\beta}^{\infty}f^{*}(x)\,dx<\infty\textrm{ if
$\sigma^{2}>2b^{\prime}(\infty)$},$
$\displaystyle\int_{\beta}^{\infty}f^{*}(x)\,dx=\infty\textrm{ if
$\sigma^{2}<2b^{\prime}(\infty)$}$ (82)

for $0<\alpha<\infty$ and $0<\beta<\infty$. From Theorem 4.6 it follows that:

  1. (i)

if $\sigma^{2}<\min\\{2b^{\prime}(0),2b^{\prime}(\infty)\\}$, then
$\lim\limits_{t\to\infty}{\rm P}(x_{t}\geq\beta)=1$,

  2. (ii)

if $\sigma^{2}>\max\\{2b^{\prime}(0),2b^{\prime}(\infty)\\}$, then
$\lim\limits_{t\to\infty}{\rm P}(x_{t}\leq\alpha)=1$,

  3. (iii)

if $2b^{\prime}(0)<\sigma^{2}<2b^{\prime}(\infty)$, then
$\limsup\limits_{t\to\infty}{\rm P}(x_{t}\leq\alpha)>0$ and
$\limsup\limits_{t\to\infty}{\rm P}(x_{t}\geq\beta)>0$.

It can be check that the limits superior in (iii) can be replaced by limits,
but their values depend on the distribution of $x_{0}$. Let
$b_{0}:=b^{\prime}(0)-\sigma^{2}/2$ and
$b_{\infty}:=b^{\prime}(\infty)-\sigma^{2}/2$. Then $b_{0}$ and $b_{\infty}$
are called the actual growth rate at $0$ and $\infty$. Now conditions
(i)–(iii) say: if both actual growth rates are positive, then the population
grows indefinitely; if both actual growth rates are negative, then the
population is becoming extinct; if $b_{0}<0$ and $b_{\infty}>0$, then the
population can with positive probability both grow indefinitely and become
extinct.

###  5.4 Gene expression

Gene expression is a process of production and decay of various biomolecules
dependent on activity of a number of genes. Changes in concentration of these
biomolecules can be basically described by a system of differential equations,
but this system depends on random parameters related to the genes’ activity.

Assume that genes can form $k+1$ configurations depending on their state of
activity and for configuration $i\in I=\\{0,\dots,k\\}$ concentration of $d$
different biomolecules changes according to a system of differential equations
of the form $x^{\prime}=b^{i}(x)$, $x\in\mathbb{R}_{+}^{d}$. We assume that
for each $x_{0}\in\mathbb{R}_{+}^{d}$ there exists a unique solution
$x\colon\mathbb{R}_{+}\to\mathbb{R}_{+}^{d}$ of the last equation with the
initial condition $x(0)=x_{0}$ and we denote this solution by
$\pi_{t}^{i}x_{0}$. Assume that for $i,j\in I$ and $i\neq j$ and for
concentration $x$, the rate of switching configuration from $i$ to $j$ is a
nonnegative continuous and bounded function $x\mapsto q_{ij}(x)$.

We start with a simple model in which protein molecules are directly
”produced” by a single gene. A gene can be in an active $i(t)=1$ or inactive
state $i(t)=0$. When gene is active then proteins are produced with a constant
speed $P$. In both states $\mu$ is the rate of protein degradation. Let $x(t)$
be the concentration of protein. Then $x(t)$ satisfies the equation

$x^{\prime}(t)=Pi(t)-\mu x(t).$ (83)

Gene expression is described by dynamical systems with random switching
$\xi_{t}=(x(t),i(t))$, $t\geq 0$, with values in the space
$\mathbb{R}_{+}\times\\{0,1\\}$ and with random jumps from $(x,0)$ to $(x,1)$
and from $(x,1)$ to $(x,0)$ with rates $q_{01}(x)$ and $q_{10}(x)$,
respectively. According to Sec. 3.6 this process generates a stochastic
semigroup on the space $L^{1}(X,\mathcal{B}(X),m)$, where
$X=\mathbb{R}_{+}\times\\{0,1\\}$ and $m$ is the product of the Lebesgue
measure on $\mathbb{R}_{+}$ and the counting measure on the set $\\{0,1\\}$.

We now present a version of this model with non-random moments of jumps. We
assume that the concentration of protein must not fall below a fixed level
$\theta>0$. We assume that the gene is inactivated with rate $q_{10}(x)$ but
if the protein concentration reaches the level $\theta$, the protein
production is automatically activated (Fig. 5).

Figure 5: Gene expression scheme with jumps on the boundary.

Now gene expression is described by a piecewise deterministic Markov process
$\xi_{t}=(x(t),i(t))$, $t\geq 0$, with values in the space
$\big{[}\theta,\frac{P}{\mu}\big{]}\times\\{0,1\\}$ with random jumps form
$(x,1)$ to $(x,0)$ and with non-random jumps from $(\theta,0)$ to
$(\theta,1)$.

If $x_{0}$ the initial number of protein in the active state, then the length
of the active state has the distribution

$F_{1}(t)=1-\exp\bigg{\\{}-\int_{0}^{t}q_{10}(1+(x_{0}-1)e^{-s})\,ds\bigg{\\}}.$
(84)

The duration of stay in the inactive state $T$ satisfies the equation
$x_{0}e^{-\mu t}=\theta$, thus $T=\frac{1}{\mu}\ln(x_{0}/\theta)$. The length
distribution of the inactive state is therefore the function

$F_{0}(t)=\begin{cases}0,&\text{if $t<T$},\\\ 1,&\text{if $t\geq
T$}.\end{cases}$ (85)

Now we consider more realistic model introduced in lipniacki2  and studied in
BLPR . A model consists of two types of molecules: mRNA, and protein, and we
assume that variables $x_{1},x_{2}$ denote their concentrations. If the gene
is active then mRNA transcript molecules are synthesized with a constant speed
$R$. The protein translation proceeds with the rate $Px_{1}(t)$, where $P$ is
a constant. The mRNA and protein degradation rates are $\mu_{R}$ and
$\mu_{P}$, respectively. Thus the process is described by the following system

$\left\\{\begin{aligned} x_{1}^{\prime}(t)&=Ri(t)-\mu_{R}x_{1}(t),\\\
x_{2}^{\prime}(t)&=Px_{1}(t)-\mu_{P}x_{2}(t),\end{aligned}\right.$ (86)

where $i(t)=1$ if the gene is active and $i(t)=0$ in the opposite case.

We assume that the rates $q_{01}$ and $q_{10}$ are continuous and bounded
functions of $\mathbf{x}=(x_{1},x_{2})$. Then the process
$\xi_{t}=(x_{1}(t),x_{2}(t),i(t))$, $t\geq 0$, is given by dynamical systems
with random switching. The trajectories of the process enter the invariant set
$X=\Big{[}0,\frac{R}{\mu_{R}}\Big{]}\times\Big{[}0,\frac{PR}{\mu_{P}\mu_{R}}\Big{]}\times\\{0,1\\}$,
so we can consider the process $(\xi_{t})_{t\geq 0}$ with values in this set.
It generates a stochastic semigroup $\\{P(t)\\}_{t\geq 0}$ on the space
$L^{1}(X,\mathcal{B}(X),m)$.

In this model we have two dynamical systems, which correspond, respectively,
to $i=0$ and $i=1$. Each system has a unique stationary point, either
$\mathbf{x}_{0}=(0,0)$, or
$\mathbf{x}_{1}=\Big{(}\frac{R}{\mu_{R}},\frac{PR}{\mu_{P}\mu_{R}}\Big{)}$,
which is asymptotically stable. We assume that $q_{01}(\mathbf{x}_{0})>0$ and
$q_{10}(\mathbf{x}_{1})>0$. This assumption implies that the trajectories of
the process visit any neighbourhood of the point $(\mathbf{x}_{0},0)$ and the
point $(\mathbf{x}_{1},1)$ for infinitely many times. It means that condition
(WI) holds. Moreover, if we assume additionally that
$q_{01}(\mathbf{x}_{1})>0$ or $q_{10}(\mathbf{x}_{0})>0$ then we can check
that condition (K) also holds (see PR-prz-apss-apdMp  for the proof based on
Hörmander’s condition). The set $X$ is compact. According to Corollary 9, the
semigroup $\\{P(t)\\}_{t\geq 0}$ is asymptotically stable.

In some cases, the process of protein production has a pre-transcriptional
step. Activation of the gene is followed by the production of RNA molecules
called pre-mRNA, which cannot directly produce a protein. Then, while still in
the cell nucleus, non-coding sequences (introns) are truncated to form mRNA
molecules, which are then transported to the cytoplasm, where they serve as
matrices in the translation process of protein production. Such a three-step
model was introduced and investigated in RT .

Let us denote by $x_{1}(t),x_{2}(t),x_{3}(t)$ the concentration of pre-mRNA,
mRNA and protein at time $t$. The change in these quantities over time can be
represented by a system of three equations

$\left\\{\begin{aligned} x_{1}^{\prime}(t)&=Ai(t)-(R+\mu_{pR})x_{1}(t),\\\
x_{2}^{\prime}(t)&=Rx_{1}(t)-\mu_{R}x_{2}(t),\\\
x_{3}^{\prime}(t)&=Px_{2}(t)-\mu_{P}x_{3}(t),\end{aligned}\right.$ (87)

where $A$ is the speed of pre-mRNA molecules synthesis when the gene is
active, $R$ is the conversion rate of pre-mRNA molecules into active mRNA,
$\mu_{pR}$ is the degradation rate of pre-mRNA molecules, and the other
constants are the same as in the previous model.

The model (86) can also be modified to a model with non-random jumps. As in
the previous model the gene is inactivated with rate $q_{10}(x_{1},x_{2})$ but
it is activated when the protein concentration $x_{2}(t)$ reaches $\theta$,
where $0<\theta\leq PR/\mu_{R}\mu_{P}$. Note that in this case it may happen
that $x_{1}(\bar{t})<\mu_{p}\theta$ at the time $\bar{t}$ of activation, and
then $x_{2}^{\prime}(\bar{t})<0$ and the protein concentration $x_{2}(t)$ will
decrease in a certain interval $(\bar{t},\bar{t}+\varepsilon)$, in particular
will fall below the level of $\theta$. It is clear that the gene should remain
active until the concentration of the protein $x_{2}(t)$ does not exceed at
least the level of $\theta$, we should therefore assume that the inactivation
intensity $q_{10}(x_{1},x_{2})$ is zero when $x_{2}\leq\theta$. The process
$\xi_{t}=(x_{1}(t),x_{2}(t),i(t))$, $t\geq 0$ is defined on the space
$X=\big{[}0,\frac{R}{\mu_{R}}\big{]}\times\big{[}0,\frac{PR}{\mu_{P}\mu_{R}}\big{]}\times\\{0,1\\}$
and has non-random jumps on the set
$S=\big{[}0,\frac{R}{\mu_{R}}\big{]}\times\\{\theta\\}\times\\{0,1\\}$. In
this case, additional jumps occur inside the phase space.

The papers HWS ; KKLL  present more advanced models describing the process of
subtiline production by the bacterium Bachillus subtilis. Although the process
plays an important role in biochemical technology we will not discuss the
model in detail, but only highlight its interesting mathematical aspects.

To survive, the bacterium produces a type of antibiotic, subtiline, which
eliminates other microorganisms competing with it. The model considered in HWS
consists of five differential equations describing the size $x_{1}$ of the B.
subtilis population, the amount of food $x_{2}$ and the concentration
$x_{3},x_{4},x_{5}$ of the three proteins involved in the production. Two of
these equations are without switching, two with switching dependent on the
state of two genes and one with switching dependent on the amount of food:

$x_{3}^{\prime}(t)=\begin{cases}-\mu x_{3}(t),&\text{if
$x_{2}(t)\geq\theta$},\\\ P_{3}-\mu x_{3}(t),&\text{if