Markov Operators on a Banach Lattice and Their Applications

Abstract

Any Markov process is defined by a stochastic kernel. By using a stochastic kernel, we can define a Markov operator on a Banach lattice. Conversely, by using a Markov operator, we can define a Markov process. There is a close relationship between a Markov process and a Markov operator. In this chapter, we show the relation between a Markov process and a Markov operator. We show some applications of Markov operators concerned with dynamical systems and we mention the Jacobs-de Leeuw-Glicksberg decomposition which is induced by a Markov operator.

1. Introduction

We shall consider a random dynamical system. Let $S:\mathrm{ℝ}\to \mathrm{ℝ}$ be a dynamical system and define a stochastic process, ${\left\{X_n\right\}}_{n\ge 0}$ by $X_{n+1}=S\left(X_n\right)+Y_n$, where $Y_0,Y_1,\cdots$ are independent random variables with values in $\mathrm{ℝ}$, each having the same density g, and $X_0$ and ${\left\{Y_n\right\}}_{n\ge 0}$ are independent. Then the stochastic process $\left\{X_n\right\}$ is a Markov process. Let $f_n$ be the density function of $X_n$ for each $n\ge 0$, and hence, we have $f_{n+1}\left(x\right)={\int }_{\mathrm{ℝ}}{f}_n\left(y\right)g\left(x-S\left(y\right)\right)\mu \left(\mathit{dy}\right)$, where $\mu$ is the Lebesgue measure on $\mathrm{ℝ}$. This equation means that every density function $f_n$ is represented by a Markov operator, $T:L^1\left(\mathrm{ℝ}\right)\to L^1\left(\mathrm{ℝ}\right)$ defined by: $Tf\left(x\right)={\int }_{\mathrm{ℝ}}f\left(y\right)g\left(x-S\left(y\right)\right)\mu \left(\mathit{dy}\right)$ as $f_n=T^n{f}_0$. In this way, there are many results about the asymptotic behavior of their distributions represented by positive linear contractions, often called Markov operators. A positive linear contraction on $L^1$ space is associated with a transition probability for a Markov process. The iteration of a positive linear contraction expresses the asymptotic behavior of a Markov process. Especially, probability density functions with distributions for some Markov processes induced by dynamical systems and noises are sometimes represented by a Markov operator. In this work, we will explain our previous work such as some relations between the Jacobs-de Leeuw-Glicksberg decomposition of semigroups and the existence of a constrictor, which attracts densities of the Markov operators on a Banach space, which are induced by stochastic difference equations. Jacobs first obtained this splitting theorem under the reflexivity assumption. De Leeuw and Glicksberg showed the splitting theorem for an abelian weakly almost periodic semigroup of bounded operators on a complex Banach space. They also showed a similar splitting theorem for a non-abelian semigroup of linear contractions in a strictly convex Banach space with a strictly convex dual space. The Jacobs-de Leeuw-Glicksberg decomposition holds for a complex Banach space. We introduce our recent results about some relations of the existence of a constrictor of a linear contractive operator between (continuous or discrete) semigroups of linear contractive operators in a complex Banach space by using our results. Furthermore, we will consider a Markov operator T on a real $L^1\left(\Omega ,\Sigma ,\mu \right)$ space, where $\left(\Omega ,\Sigma ,\mu \right)$ is a probability measure space. The iteration of a positive linear contraction expresses the asymptotic behavior of a Markov process. Especially, probability density functions with distributions for some Markov processes induced by dynamical systems and noises are sometimes represented by a Markov operator. We study their asymptotic phenomena by using mean ergodic theorems and pointwise ergodic theorems. In this work, we study our recent operator theoretic approaches for Markov processes and their applications.

2. An operator theoretic approach to a Markov process

A stochastic kernel is a map that in the general theory of Markov processes plays the role that the transition matrix does in the theory of Markov processes with a finite state space. We shall give the basic definition concerning a stochastic kernel (see [1, 2, 3, 4, 5, 6]).

Definition 1. A stochastic kernel is any map $K:E\times \mathcal{E}\to \left[0,1\right]$ satisfying the following three conditions:

1. for any fixed set $A\in \mathcal{E}$, the function $K\left(\cdot ,A\right)$ is measurable;

2. for any fixed state $x\in E$, the set function $K\left(x,\cdot \right)$ is a measure on $\left(E,\mathcal{E}\right)$;

3. for all $x\in E,K\left(x,E\right)=1.$

Firstly, we show that any positive contraction on $L^1$-space induces the stochastic kernel. Let E be a nonempty set, $\mathcal{E}$ a $\sigma$-algebra of subsets of E, and $\left(E,\mathcal{E},m\right)$6a $\sigma$-finite measure space. Let $L^1\left(E,\mathcal{E},m\right)=L^1$ be the space of real valued integrable functions with respect to $\left(E,\mathcal{E},m\right)$ and $⟨x^{\ast },x⟩$be a dual pair of $x^{\ast }\in L^{\infty }$ and $x\in L^1$.

Definition 2. When a linear operator, T on $L^1$, satisfies the following conditions, we call T a positive contraction:

1. if $\parallel T\parallel \le 1$;

2. for any positive function $f\ge 0$ a.e., $Tf\ge 0$ a.e.

Theorem 1. Let T be a positive contraction on $L^1$. For any $A\in \mathcal{E}$, a map

$$K\left(x,A\right)=T^{\ast }{\mathbf{1}}_A\left(x\right)$$E1

satisfies the following conditions, where $T^{\ast }$ is the adjoint operator of T, and ${\mathbf{1}}_A$ is the indicator function of A:

1. for any fixed set $A\in \mathcal{E}$, the function $K\left(\cdot ,A\right)$ is measurable;

2. for almost every $x\in E$, the set function $K\left(x,\cdot \right)$ is a measure on $\left(E,\mathcal{E}\right)$;

3. for almost every $x\in E,K\left(x,E\right)\le 1.$

Proof.

1. For any $A\in \mathcal{E},T^{\ast }{\mathbf{1}}_A\in L^{\infty }\left(E,\mathcal{E},m\right);$

2. It is obvious that $K\left(x,Ø\right)=T^{\ast }{\mathbf{1}}_{Ø}=0$ a.e. since $T^{\ast }$ is a positive contraction on $L^{\infty },K\left(x,A\right)=T^{\ast }{\mathbf{1}}_A\left(x\right)\ge 0$ a.e. for any $A\in \mathcal{E}$.

   Let $A={\cup }_{i=1}^{\infty }{A}_i$ for pairwise disjoint $A_i\in \mathcal{E}$. For any $u\in L^1,u\ge 0,$

   $$\begin{array}{l}⟨T^{\ast }{\mathbf{1}}_A,u⟩=⟨{\mathbf{1}}_A,\mathit{Tu}⟩\\ =⟨{\mathbf{1}}_{{\cup }_{i=1}^{\infty }{A}_i},\mathit{Tu}⟩={\int }_{{\cup }_{i=1}^{\infty }{A}_i}\mathit{Tu}\left(x\right)m\left(\mathit{dx}\right)\\ =\sum _{i=1}^{\infty }{\int }_{A_i}\mathit{Tu}\left(x\right)m\left(\mathit{dx}\right)=\sum _{i=1}^{\infty }⟨{\mathbf{1}}_{A_i},\mathit{Tu}⟩=\sum _{i=1}^{\infty }⟨T^{\ast }{\mathbf{1}}_{A_i},u⟩\\ =\sum _{i=1}^{\infty }{\int }_{E}u\left(x\right)\cdot T^{\ast }{\mathbf{1}}_{A_i}\left(x\right)m\left(\mathit{dx}\right)=\underset{n\to \infty }{lim}{\int }_E\sum _{i=1}^{n}u\left(x\right)\cdot T^{\ast }{\mathbf{1}}_{A_i}\left(x\right)m\left(\mathit{dx}\right)\\ =\underset{n\to \infty }{lim}{\int }_{E}u\left(x\right)\cdot \sum _{i=1}^n{T}^{\ast }{\mathbf{1}}_{A_i}\left(x\right)m\left(\mathit{dx}\right)\end{array}$$E2

   Since $u,T^{\ast }{\mathbf{1}}_{A_i}\ge 0,u\cdot {\sum }_{i=1}^n{T}^{\ast }{\mathbf{1}}_{A_i}\uparrow u\cdot {\sum }_{i=1}^{\infty }{T}^{\ast }{\mathbf{1}}_{A_i}$ a.e. as $n\to \infty$. We have

   $$\underset{n\to \infty }{lim}{\int }_{E}u\left(x\right)\cdot \sum _{i=1}^n{T}^{\ast }{\mathbf{1}}_{A_i}\left(x\right)m\left(\mathit{dx}\right)={\int }_{E}u\left(x\right)\cdot \sum _{i=1}^{\infty }{T}^{\ast }{\mathbf{1}}_{A_i}\left(x\right)m\left(\mathit{dx}\right)={\int }_{E}u\left(x\right)\cdot T^{\ast }{\mathbf{1}}_A\left(x\right)m\left(\mathit{dx}\right)$$E3

   by the monotone convergence theorem. This means that

   $$⟨T^{\ast }{\mathbf{1}}_A,u⟩=⟨\sum _{i=1}^{\infty }{T}^{\ast }{\mathbf{1}}_{A_i},u⟩$$E4

   for any $u\in L^1,u\ge 0.$ Since every $u\in L^1$ is represented by $u=u^{+}-u^{-},u^{+},u^{-}\in L^1,u^{+},u^{-}\ge 0,$ we have

   $$\begin{array}{l}⟨T^{\ast }{\mathbf{1}}_A,u⟩=⟨T^{\ast }{\mathbf{1}}_A,u^{+}⟩-⟨T^{\ast }{\mathbf{1}}_A,u^{-}⟩\\ =⟨\sum _{i=1}^{\infty }{T}^{\ast }{\mathbf{1}}_{A_i},u^{+}⟩-⟨\sum _{i=1}^{\infty }{T}^{\ast }{\mathbf{1}}_{A_i},u^{-}⟩\\ =⟨\sum _{i=1}^{\infty }{T}^{\ast }{\mathbf{1}}_{A_i},u⟩.\end{array}$$E5

   Therefore, $K\left(x,\cdot \right)$ is completely additive, that is, if $A={\bigcup }_{i=1}^{\infty }{A}_i$ for disjoint $A_i\in \mathcal{E}$, then

   $$K\left(x,A\right)=\sum _{i=1}^{\infty }K\left(x,A_i\right)\text{for}\mathrm{a}.\mathrm{e}.x.$$E6

3. Since $T^{\ast }$ is a positive contraction on $L^{\infty },K\left(x,E\right)=T^{\ast }{\mathbf{1}}_E\left(x\right)\le {\mathbf{1}}_E\left(x\right)$ a.e.

Remark 1. If the results 1–3 of theorem 1 hold for everywhere $x\in E$, then K(x, A) is stochastic kernel (see Definition 1); see [7, 8] for more detail.

Secondly, we show that any stochastic kernel induces the positive contraction on $L^1$-space.

Theorem 2. Let $K:E\times \mathcal{E}\to \left[0,1\right]$ be a stochastic kernel satisfying the condition

$$K\left(x,A\right)=0a.e.x\in E$$E7

for any $A\in \mathcal{E},m\left(A\right)=0.$ Then, for any absolutely continuous measure $\mu$ with respect to m, the measure $\nu$ on $\mathcal{E}$:

$$\nu \left(A\right)={\int }_{E}K\left(x,A\right)\mu \left(\mathit{dx}\right)A\in \mathcal{E}$$E8

is an absolutely continuous measure with respect to m, and there exists the positive contraction T on $L^1$ such that

$$\tilde{\nu }=T\tilde{\mu },$$E9

where $\tilde{\mu },\tilde{\nu }\in L^1$ are the Radon-Nikodym derivatives of $\mu ,\nu$, respectively.

Proof. We show that for any absolutely continuous measure $\mu$ with respect to m,

$$\nu \left(A\right)={\int }_{E}K\left(x,A\right)\mu \left(\mathit{dx}\right)$$E10

is an absolutely continuous measure with respect to m.

From the definitions 1 and 2 of a stochastic kernel, we have $\nu \left(Ø\right)=0$ and $\nu \left(A\right)\ge 0$ for any $A\in \mathcal{E}$.

Let $A={\cup }_{i=1}^{\infty }{A}_i$ for pairwise disjoint $A_i\in \mathcal{E}$. Since $K\left(x,A_i\right)\ge 0$, it follows that

$$\begin{array}{l}\nu \left(A\right)={\int }_{E}K\left(x,A\right)\mu \left(\mathit{dx}\right)={\int }_{E}K\left(x,{\cup }_{i=1}^{\infty }{A}_i\right)\mu \left(\mathit{dx}\right)\\ ={\int }_E\sum _{i=1}^{\infty }K\left(x,A_i\right)\mu \left(\mathit{dx}\right)=\sum _{i=1}^{\infty }{\int }_{E}K\left(x,A_i\right)\mu \left(\mathit{dx}\right)=\sum _{i=1}^{\infty }\nu \left(A_i\right)\end{array}$$E11

by the monotone convergence theorem: hence, $\nu$ is a measure on $\mathcal{E}$.

From the condition, $K\left(x,A\right)=0$ for any $A\in \mathcal{E},m\left(A\right)=0,$; thus, we have

$$\nu \left(A\right)={\int }_{E}K\left(x,A\right)\mu \left(\mathit{dx}\right)={\int }_{E}0\mu \left(\mathit{dx}\right)=0,$$E12

which means that $\nu$ is an absolutely continuous measure with respect to m.

Since m is a $\sigma$-finite measure, by the Radon-Nikodym theorem, there are the Radon-Nikodym derivatives $\tilde{\mu },\tilde{\nu }\in L^1,\tilde{\mu },\tilde{\nu }\ge 0$ such as

$$\mu \left(A\right)={\int }_A\tilde{\mu }\left(x\right)m\left(\mathit{dx}\right)\text{and}\nu \left(A\right)={\int }_A\tilde{\nu }\left(x\right)m\left(\mathit{dx}\right)$$E13

for any $A\in \mathcal{E}$. We can define an operator T on the positive cone $L_{+}^1$ of $L^1$ such that

$$\tilde{\nu }=T\tilde{\mu }.$$E14

For notational convenience, we shall write $\mathit{T\mu }$ for $\nu$.

For any $f\in L^1$, the completely additive set function

$$\phi \left(A\right)={\int }_{A}f\left(x\right)m\left(\mathit{dx}\right)A\in \mathcal{E}$$E15

is absolutely continuous with respect to m. Let $\phi ={\phi }_{+}-{\phi }_{-}$ be the Jordan decomposition of $\phi$. Both ${\phi }_{+}$ and ${\phi }_{-}$ are absolutely continuous with respect to m; then there exist the Radon-Nikodym derivatives $f_{+},f_{-}\in L_{+}^1$ of ${\phi }_{+},{\phi }_{-}$, respectively and $f\left(x\right)=f_{+}\left(x\right)-f_{-}\left(x\right)$ a.e. $x\in E$. By using them, we define $\mathit{T\phi }$ as follows:

$$\begin{array}{l}\mathit{T\phi }={\int }_{E}K\left(x,A\right)\phi \left(\mathit{dx}\right)={\int }_{E}K\left(x,A\right)f\left(x\right)m\left(\mathit{dx}\right)\\ ={\int }_{E}K\left(x,A\right)\left(f_{+}\left(x\right)-f_{-}\left(x\right)\right)m\left(\mathit{dx}\right)={\int }_{E}K\left(x,A\right)f_{+}\left(x\right)m\left(\mathit{dx}\right)-{\int }_{E}K\left(x,A\right)f_{-}\left(x\right)m\left(\mathit{dx}\right)\\ ={\int }_{E}K\left(x,A\right){\phi }_{+}\left(\mathit{dx}\right)-{\int }_{E}K\left(x,A\right){\phi }_{-}m\left(\mathit{dx}\right)=T{\phi }_{+}-T{\phi }_{-}.\end{array}$$E16

$\mathit{T\phi }$ is also absolutely continuous with respect to m. Thus, there exist the Radon-Nikodym derivatives $g\in L^1$ of $\mathit{T\phi }$, and we define $Tf=g$. From the definition of T, T is a positive linear operator on $L^1$.

Finally, we prove that T is a contraction on $L^1$. For any $f\in L^1$, we obtain

$$\parallel f\parallel ={\int }_E{f}_{+}\left(x\right)m\left(\mathit{dx}\right)+{\int }_E{f}_{-}\left(x\right)m\left(\mathit{dx}\right)={\phi }_{+}\left(E\right)+{\phi }_{-}\left(E\right).$$E17

In a similar way, we can show that

$$\parallel Tf\parallel =T{\phi }_{+}\left(E\right)+T{\phi }_{-}\left(E\right).$$E18

Then we have

$$\begin{array}{l}T{\phi }_{+}\left(E\right)={\int }_{E}K\left(x,A\right){\phi }_{+}\left(\mathit{dx}\right)\le {\int }_E{\mathbf{1}}_E\left(x\right){\phi }_{+}\left(\mathit{dx}\right)={\phi }_{+}\left(E\right),\\ T{\phi }_{-}\left(E\right)={\int }_{E}K\left(x,A\right){\phi }_{-}\left(\mathit{dx}\right)\le {\int }_E{\mathbf{1}}_E\left(x\right){\phi }_{-}\left(\mathit{dx}\right)={\phi }_{-}\left(E\right).\end{array}$$E19

This implies that $\parallel Tf\parallel \le \parallel f\parallel$.

We remark that the condition (7) in Theorem 2 is necessary.

Lemma 1 ([8]). A stochastic kernel $K:E\times \mathcal{E}\to \left[0,1\right]$ satisfies the condition (7) if and only if for any absolutely continuous measure $\mu$ with respect to m,

$$\nu \left(A\right)={\int }_{E}K\left(x,A\right)\mu \left(\mathit{dx}\right)$$E20

is an absolutely continuous measure with respect to m.

Proof. If K does not satisfy (7), then there exist a set, $A\in \mathcal{E}$, and an element, $x\in E$, such that $m\left(A\right)=0$, and $K\left(x,A\right)>0$. We define a set, $B=\left\{x\in E:K\left(x,A\right)>0\right\}\in \mathcal{E}$. B is not empty, and $m\left(B\right)>0$. The measure $\mu$ on $\mathcal{E}$ defined by $\mu \left(C\right)=m\left(C\cap B\right)$ for any $C\in \mathcal{E}$ is absolutely continuous with respect to m, but $\nu \left(A\right)={\int }_{E}K\left(x,A\right)\mu \left(\mathit{dx}\right)={\int }_{B}K\left(x,A\right)m\left(\mathit{dx}\right)>0$.

3. Markov operators induced by a dynamical system

In the previous section, we showed the relations between a stochastic kernel and a positive contraction. In this section, we shall treat a certain kind of positive contraction, which is called a Markov operator.

Firstly, we give the definition of a Markov operator on $L^1$-space (see [9, 10, 11]). Let E be a nonempty set and $\mathcal{E}$ a $\sigma$-algebra of subsets of E. Let $\left(E,\mathcal{E},m\right)$ be a measure space, $L^1\left(E,\mathcal{E},m\right)$ be the space of real valued integrable functions with respect to $\left(E,\mathcal{E},m\right)$ and $⟨x^{\ast },x⟩$be a dual pair of $x^{\ast }\in L^{\infty }$ and $x\in L^1$.

Definition 3. A liner operator T in an $L^1\left(E,\mathcal{E},m\right)$ is called a Markov operator if

$$Tf\ge 0\text{and}\parallel Tf\parallel =\parallel f\parallel \text{for}\mathrm{all}f\in L_{+}^1\left(E,\mathcal{E},m\right),$$E21

where $\parallel \cdot \parallel$ is the $L^1\left(E,\mathcal{E},m\right)$-norm, and $L_{+}^1$ is the positive cone of $L^1\left(E,\mathcal{E},m\right)$.

From this definition, obviously, Markov operators are positive operators and satisfy the following properties.

Preposition 1. Markov operators are linear contraction in $L^1$-space.

Proof. Since T is a positive operator, we have that for m-almost everywhere $x\in E$

$$\begin{array}{l}{\left(Tf\right)}^{+}\left(x\right)={\left(T{f}^{+}\left(x\right)-T{f}^{-}\left(x\right)\right)}^{+}=max\left\{0,T{f}^{+}\left(x\right)-T{f}^{-}\left(x\right)\right\}\\ \le max\left\{0,T{f}^{+}\left(x\right)\right\}=T{f}^{+}\left(x\right);\\ {\left(Tf\right)}^{-}\left(x\right)={\left(T{f}^{+}\left(x\right)-T{f}^{-}\left(x\right)\right)}^{-}=max\left\{0,T{f}^{-}\left(x\right)-T{f}^{+}\left(x\right)\right\}\\ \le max\left\{0,T{f}^{-}\left(x\right)\right\}=T{f}^{-}\left(x\right),\end{array}$$E22

where $f^{+}\left(x\right)=max\left\{0,f\left(x\right)\right\}$ and $f^{-}\left(x\right)=max\left\{0,-f\left(x\right)\right\}$. Thus, it follows that

$$\mid Tf\mid ={\left(Tf\right)}^{+}+{\left(Tf\right)}^{-}\le T{f}^{+}+T{f}^{-}=T\left(f^{+}+f^{-}\right)=T\mid f\mid \mathrm{a}.\mathrm{e}.$$E23

From this inequality and the second condition of the definition of Markov operator, we have that

$$\parallel Tf\parallel ={\int }_E\mid Tf\left(x\right)\mid m\left(\mathit{dx}\right)\le {\int }_{E}T\mid f\left(x\right)\mid m\left(\mathit{dx}\right)={\int }_E\mid f\mid m\left(\mathit{dx}\right)=\parallel f\parallel .$$E24

This implies that T is a contraction in $L^1$-space.

Preposition 2. The second condition of Markov operator: that is, $\parallel Tf\parallel =\parallel f\parallel$ for $f\in L_{+}^1$, is equivalently expressed as $T^{\ast }{\mathbf{1}}_E={\mathbf{1}}_E$, where $T^{\ast }$ is the adjoint operator of T, and ${\mathbf{1}}_E\left(x\right)$ is the indicator function of E.

Proof. By the second condition of Markov operator, we have for any $C\in \mathcal{E}$

$${\int }_C{T}^{\ast }{\mathbf{1}}_E\left(x\right)m\left(\mathit{dx}\right)=⟨T^{\ast }{\mathbf{1}}_E,{\mathbf{1}}_C⟩=⟨{\mathbf{1}}_E,T{\mathbf{1}}_C⟩=⟨{\mathbf{1}}_E,{\mathbf{1}}_C⟩={\int }_C{\mathbf{1}}_E\left(x\right)m\left(\mathit{dx}\right).$$E25

This implies that $T^{\ast }{\mathbf{1}}_E={\mathbf{1}}_E$.

Conversely, if $T^{\ast }{\mathbf{1}}_E={\mathbf{1}}_E$, then we have for $f\in L_{+}^1$

$$\parallel Tf\parallel =⟨{\mathbf{1}}_E,|Tf|⟩=⟨{\mathbf{1}}_E,Tf⟩=⟨T^{\ast }{\mathbf{1}}_E,f⟩=⟨{\mathbf{1}}_E,f⟩=\parallel f\parallel .$$E26

Remark 2. Notice that sometimes a Markov operator is considered as a positive linear contraction defined on $L^1$-space because of Theorem 2, and there are many important results about positive linear contraction in the Ergodic theory. For example, the splitting theorem of Jacobs-Deleeuw-Glicksberg (see [8]) and “zero-two”law by Ornstein and Sucheston (see [12], 1970).

Next, we consider a measurable function, $K:E\times E\to \mathbb{R}$ such that

$$K\left(x,y\right)\ge 0\text{for}\mathrm{all}x,y\in E$$E27

and

$${\int }_{E}K\left(x,y\right)m\left(\mathit{dx}\right)=1\text{for}\mathrm{a}.\mathrm{e}.y\in E.$$E28

By (27) and (28),

$$\begin{array}{l}{\int }_E\left({\int }_E|K\left(x,y\right)f\left(y\right)|m\left(\mathit{dx}\right)\right)m\left(\mathit{dy}\right)={\int }_E\left({\int }_{E}K\left(x,y\right)m\left(\mathit{dx}\right)\right)\mid f\left(y\right)\mid m\left(\mathit{dy}\right)\\ ={\int }_E\mid f\left(y\right)\mid m\left(\mathit{dy}\right)=\parallel f\parallel \end{array}$$E29

for any $f\in L^1$; hence, $\mid K\left(x,y\right)f\left(y\right)\mid$ is integrable on $E\times E$ and

$$\begin{array}{l}\parallel f\parallel ={\int }_E\left({\int }_E|K\left(x,y\right)f\left(y\right)|m\left(\mathit{dx}\right)\right)m\left(\mathit{dy}\right)={\int }_E\left({\int }_E|K\left(x,y\right)f\left(y\right)|m\left(\mathit{dy}\right)\right)m\left(\mathit{dx}\right)\\ \ge {\int }_E\left|{\int }_{E}K\left(x,y\right)f\left(y\right)m\left(\mathit{dy}\right)\right|m\left(\mathit{dx}\right)\end{array}$$E30

by Fubini’s Theorem (see). Thus, we can define an integral operator T on $L^1$ by

$$Tf\left(x\right)={\int }_{E}f\left(y\right)K\left(x,y\right)m\left(\mathit{dy}\right).$$E31

We have

$$\parallel f\parallel \ge \parallel Tf\parallel .$$E32

Then, T is a positive contraction on $L^1$. Furthermore, T is a Markov operator on $L^1$. Indeed, we have that for any $f\in L_{+}^1$,

$$\begin{array}{l}\parallel Tf\parallel ={\int }_{E}Tf\left(x\right)m\left(\mathit{dx}\right)={\int }_E\left({\int }_{E}f\left(y\right)K(x,y)m\left(\mathit{dy}\right)\right)m\left(\mathit{dx}\right)\\ ={\int }_E\left({\int }_{E}K\left(x,y\right)m\left(\mathit{dx}\right)\right)f\left(y\right)m\left(\mathit{dy}\right)={\int }_{E}f\left(y\right)m\left(\mathit{dy}\right)\\ =\parallel f\parallel .\end{array}$$E33

For the sake of convenience and without loss of generality, we shall assume that E is $\mathbb{R},\mathcal{E}$ the $\sigma$-algebra of Borel subsets, and $\lambda$ the Lebesgue measure on $\mathbb{R}$. We shall give some stochastic processes, which induce Markov operators as an integral operator defined by (31) for a given measurable function K(x, y).

Let $\left(\Omega ,\mathcal{F},P\right)$ be a probability space, where $\mathcal{F}$ denotes a Borel $\sigma$-field and P a probability measure, $X_0,Y_0,Y_1,\cdots$ independent $\mathbb{R}$-valued random variables with densities $f_0$ and $g_n=g$, respectively, and $S:\mathbb{R}\to \mathbb{R}$ be a measurable transformation. Suppose that $g\in D≔\left\{f\in L^1:\parallel f\parallel =1,f\ge 0\right\}$. Under these conditions, we shall give examples of two types of random dynamical systems, for which Markov operators are induced.

• Additive noise type: Consider the following stochastic process $\left\{X_n\right\}$ defined by

$$X_{n+1}=S\left(X_n\right)+Y_n$$E34

for each $n\ge 0$.

Since $X_0$ and $\left\{Y_n\right\}$ are independent and $f_n,g$ are integrable, we have that

$$\begin{array}{l}P\left(X_{n+1}\in A\right)=P\left(S\left(X_n\right)+Y_n\in A\right)=\int {\int }_{S\left(x\right)+y\in A}{f}_n\left(y\right)g\left(x\right)\lambda \left(\mathit{dy}\right)\lambda \left(\mathit{dx}\right)\\ ={\int }_{z\in A}\left\{{\int }_{y\in \mathbb{R}}{f}_n\left(y\right)g\left(z-S\left(y\right)\right)\lambda \left(\mathit{dy}\right)\right\}\lambda \left(\mathit{dz}\right)\\ ={\int }_A{T}^n{f}_0\left(z\right)\lambda \left(\mathit{dz}\right).\end{array}$$E35

If we set $K\left(x,y\right)=g\left(x-S\left(y\right)\right)$, then K(x, y) satisfies (27) and (28). Thus, the density function $f_n$ of $X_n$ is represented by the n-th iteration of the linear operator $T:L^1\left(\mathbb{R}\right)\to L^1\left(\mathbb{R}\right)$. Now we show that T is a Markov operator. Since $g\ge 0$ and $\parallel g\parallel =1$, we have that for any $f\in L^1\left(\mathbb{R}\right)$ with $f\ge 0$

$$\begin{array}{l}\parallel Tf\parallel ={\int }_{\mathbb{R}}\left\{{\int }_{\mathbb{R}}f\left(y\right)g\left(x-S\left(y\right)\right)\lambda \left(\mathit{dy}\right)\right\}\lambda \left(\mathit{dx}\right)\\ ={\int }_{\mathbb{R}}f\left(y\right)\left\{{\int }_{\mathbb{R}}g\left(x-S\left(y\right)\right)\lambda \left(\mathit{dx}\right)\right\}\lambda \left(\mathit{dy}\right)\\ ={\int }_{\mathbb{R}}f\left(y\right)\lambda \left(\mathit{dy}\right)=\parallel f\parallel \end{array}$$E36

by the Fubini’s theorem. Therefore, $T:L^1\left(\mathbb{R}\right)\to L^1\left(\mathbb{R}\right)$ is a Markov operator.

• Example [13]: Let $X_0,Y_0,Y_1,\cdots$ be independent $\mathbb{R}$-valued random variables with densities $f_0$ and $g_n=g\in D$, respectively, and $S:\mathbb{R}\to \mathbb{R}$ be a Borel measurable transformation, satisfying

$$\mid S\left(x\right)\mid \le \alpha \mid x\mid +\beta \text{for}x\in \mathbb{R},$$E37

where $0<\alpha <1$ and $\beta >0$.

We assume that the density g of $Y_n$ has a finite first moment, that is,

$${\int }_{\mathbb{R}}\mid x\mid g\left(x\right)\lambda \left(\mathit{dx}\right)<\infty .$$E38

Under these conditions, we consider the following stochastic process $\left\{X_n\right\}$, that is, $\left\{X_n\right\}$ is the dynamical system perturbed by additive noises:

$$X_{n+1}=S\left(X_n\right)+Y_n\text{for}n=0,1,\dots .$$E39

In this case, the Markov operator $T:L^1\left(\mathbb{R}\right)\to L^1\left(\mathbb{R}\right)$ defined by

$$Tf\left(x\right)={\int }_{\mathbb{R}}f\left(y\right)g\left(x-S\left(y\right)\right)\lambda \left(\mathit{dy}\right)$$E40

is weakly constrictive, that is, there exists the weakly precompact set $\mathcal{F}\subset L^1\left(\mathbb{R}\right)$ such that

$$\underset{n\to \infty }{lim}d\left(T^{n}f,\mathcal{F}\right)=0$$E41

for $f\in D$, where $d\left(f,\mathcal{F}\right)$ denotes the distance with respect to the norm $\parallel \cdot \parallel$ between f and the set $\mathcal{F}$. This implies that $L^1\left(\mathbb{R}\right)=E_{\mathit{fl}}\left(T\right)\oplus E_{\mathit{rev}}\left(T\right)$ with

$$E_{\mathit{fl}}\left(T\right)=\left\{x\in X:\underset{n\to \infty }{lim}\parallel T^{n}x\parallel =0\right\}\text{and}dim\left(E_{\mathit{rev}}\left(T\right)\right)<\infty .$$E42

Furthermore, Lasota and Mackey show numerically that the limiting densities of $\left\{X_n\right\}$ become asymptotically periodic with period 3 [13], and refer to Section 3 for more details.

• Multiplicative noise type: Consider the following stochastic process ${\left\{X_n^{\epsilon }\right\}}_{n\ge 0}$ defined by

$$X_{n+1}^{\epsilon }\left(\omega \right)=\left(1-\epsilon Y_n\right)S\left(X_n^{\epsilon }\left(\omega \right)\right)\text{for}\mathrm{all}n\ge 0,$$E43

where $X_0^{\epsilon }=X_0$ for each $0<\epsilon <1$.

By a similar argument as additive noise, the density function $f_n^{\epsilon }$ of $X_n^{\epsilon }$ is represented by the n-th iteration of the linear operator $T_{\epsilon }$ because $X_0$ and $\left\{Y_n\right\}$ are independent. Actually, if we assume that there exists the density function $f_n^{\epsilon }$ of $X_n^{\epsilon }$, then we have the following formula for any Borel subset $A\subset \mathbb{R}$:

$$\begin{array}{l}P\left(X_{n+1}^{\epsilon }\in A\right)=\int {\int }_{\mathit{xS}\left(y\right)\in A}{f}_n^{\epsilon }\left(y\right)\frac{1}{\epsilon }g\left(\frac{1-x}{\epsilon }\right)\lambda \left(\mathit{dy}\right)\lambda \left(\mathit{dx}\right)\\ ={\int }_A\left\{{\int }_{\mathbb{R}}{f}_n^{\epsilon }\left(y\right)g\left(\frac{1}{\epsilon }\left(1-\frac{x}{S\left(y\right)}\right)\right)\frac{1}{\mathit{\epsilon S}\left(y\right)}\lambda \left(\mathit{dy}\right)\right\}\lambda \left(\mathit{dx}\right)\\ ={\int }_A{T}_{\epsilon }{f}_n^{\epsilon }\left(x\right)\lambda \left(\mathit{dx}\right).\end{array}$$E44

This implies that $P\left(X_{n+1}^{\epsilon }\in A\right)={\int }_A{T}_{\epsilon }^{n+1}{f}_0\left(x\right)\lambda \left(\mathit{dx}\right)$. Moreover, it is easy to see that $T_{\epsilon }$ is the Markov operator, that is, $T_{\epsilon }f\ge 0$ and $\parallel T_{\epsilon }f\parallel =\parallel f\parallel$ for any $f\in L^1\left(\left[0,1\right]\right)$ with $f\ge 0$ because

$$\begin{array}{l}\parallel T_{\epsilon }f\parallel ={\int }_{\left[0,1\right]}f\left(y\right)\left\{{\int }_{\left[\frac{1}{\epsilon }-\frac{1}{\mathit{\epsilon S}\left(y\right)},\frac{1}{\epsilon }\right]\cap \left[0,1\right]}g\left(x\right)\lambda \left(\mathit{dx}\right)\right\}\lambda \left(\mathit{dy}\right)={\int }_{\left[0,1\right]}f\left(y\right)\lambda \left(\mathit{dy}\right)\\ =\parallel f\parallel \end{array}$$E45

by Fubini’s theorem (see [14] for more details). In this case, we can study how the stochastic process $\left\{X_n^{\epsilon }\right\}$ is changed as $\epsilon \to 0$.

• Example [14]: Let $S:\left[0,1\right]\to \left[0,1\right]$ be a non-singular transformation and ${\left\{Y_n\right\}}_{n\ge 0}$ be an i.i.d. sequence. We assume that the density function g for ${\left\{Y_n\right\}}_{n\ge 0}$ has full support in [0, 1], and the following conditions are satisfied:

  1. There exists a partition $0=a_1<a_2<\cdots <a_m=1$ such that for each integer $j=1,\cdots ,m-1$, the restriction $S_j$ of S to the interval $\left[a_j,a_{j+1}\right)$ is a monotone $C^1$-function and

     $$c_j≔\underset{x\in \left[a_j,a_{j+1}\right)}{inf}\left\{|S_j^{\prime }\left(x\right)|\right\}>0,c_{m-1}≔\underset{x\in \left[a_{m-1},a_m\right]}{inf}\left\{|S_j^{\prime }\left(x\right)|\right\}>0$$E46

     for $j=1,\cdots ,m-2$.

  2. $g\in L^{\infty }\left(\mathbb{R}\right);$

  3. $c_1>\frac{1}{1-{\epsilon }_0}$ for some ${\epsilon }_0\in \left(0,1\right)$.

Under these conditions, we consider the stochastic process $\left\{X_n^{\epsilon }\right\}$ defined by (43), where $\epsilon <{\epsilon }_0$. The density functions of $\left\{X_n^{\epsilon }\right\}$ are represented by the Markov operator

$$T_{\epsilon }f\left(x\right)={\int }_{\left[0,1\right]\backslash \left\{0\right\}}{P}_{S}f\left(y\right)g\left(\frac{1}{\epsilon }\left(1-\frac{x}{y}\right)\right)\frac{1}{\mathit{\epsilon y}}\lambda \left(\mathit{dy}\right).$$E47

In this case, the Markov operator $T_{\epsilon }$ is weakly constrictive for any $\epsilon <{\epsilon }_0$. Moreover, if $g\left(x\right)>0$ for all $x\in \left[0,1\right]$, then the Markov operators $T_{\epsilon }$ is asymptotically stable; that is, there is a unique $f_{\ast }^{\epsilon }\in D$ such that $T_{\epsilon }{f}_{\ast }^{\epsilon }=f_{\ast }^{\epsilon }$ and

$$\underset{n\to \infty }{lim}\parallel T_{\epsilon }^{n}f-f_{\ast }^{\epsilon }{\parallel }_1=0\text{for}\mathrm{all}f\in D,$$E48

where $D=\left\{f\in L^1\left(\mathbb{R}\right)|f\ge 0,\parallel f\parallel =1\right\}$.

Remark 3. If $S:\mathbb{R}\to \mathbb{R}$ is a non-singular measurable transformation (i.e., $\lambda \left(S^{-1}\left(A\right)\right)=0$ for any Borel set $A\subset \mathbb{R}$ with $\lambda \left(A\right)=0$), then there exists a linear operator, $T_S:L^1\left(\mathbb{R}\right)\to L^1\left(\mathbb{R}\right)$, defined by

$${\int }_A{T}_S{f}_0\left(x\right)\lambda \left(\mathit{dx}\right)={\int }_{S^{-1}\left(A\right)}{f}_0\left(x\right)\lambda \left(\mathit{dx}\right)\text{for}\text{Borel}\mathrm{set}A\subset \mathbb{R},$$E49

and called the Perron-Frobenius operator corresponding to S.

4. The iteration of a linear contractive operator on Banach space

The iteration of a positive linear contraction expresses the asymptotic behavior of a Markov process. Let E be a real or complex Banach space and T be a contractive linear operator on E. If E can be decomposed into the direct sum

$$E=E_{\mathit{fl}}\left(T\right)\oplus E_{\mathit{rev}}\left(T\right)$$E50

with respect to T, where

$$\begin{array}{l}{E}_{\mathit{fl}}\left(T\right)≔\left\{x\in E:0\in \mathrm{w}‐\mathrm{cl}\left\{T^{n}x,n\in \mathbb{N}\cup \left\{0\right\}\right\}\right\},\text{and}\\ E_{\mathit{rev}}\left(T\right)≔\left\{x\in E:y\in \mathrm{w}‐\mathrm{cl}\left\{T^{n}x,n\in \mathbb{N}\cup \left\{0\right\}\right\}\Rightarrow x\in \mathrm{w}‐\mathrm{cl}\left\{T^{n}y,n\in \mathbb{N}\cup \left\{0\right\}\right\}\right\},\end{array}$$E51

then we call this decomposition the Jacobs-de Leeuw-Glicksberg decomposition (see [8, 10, 15]). This decomposition plays a very important role in this section.

Definition 4 [16]. We call a linear contraction T on a Banach space $\left(E,\parallel \cdot \parallel \right)$ constrictive if there is a compact subset, $A\subset E$ such that

$$\underset{n\to \infty }{lim}\underset{y\in A}{inf}\parallel T^{n}x-y\parallel =0\text{for}\text{each}x\in B\left(E\right),$$E52

where B(E) is the closed unit ball of E. We call A a constrictor for T.

Theorem 3 [17]. Let E be a strictly convex and reflexive complex Banach space with the strictly convex dual space $E^{\ast }$. Given a linear contractive operator T on a complex Banach space E, the following assertions are equivalent:

1. T is constrictive.

2. $x\in E_{\mathit{fl}}\left(T\right)=\left\{x\in X:{lim}_{n\to \infty }\parallel T^{n}x\parallel =0\right\}$ if and only if $⟨x,h^{\ast }⟩=0$ holds for all eigenvectors $h^{\ast }$ of $T^{\ast }$ having unimodular eigenvalues.

Theorem 4 [17]. Let $\left(\Omega ,\Sigma ,\mu \right)$ be a finite measure space and T be a Markov operator on real $L^1\left(\Omega \right)$ defined by (31), where $K:\Omega \times \Omega \to \mathbb{R}$ is a measurable function that satisfies (27) and (28). Consider the following assertions:

1. T satisfies $\parallel Tx{\parallel }_p\le \parallel x{\parallel }_p$ for some $1\le p<\infty$, where $\parallel \cdot {\parallel }_p$ denotes the $L^p$-norm.

2. The sub $\sigma$-algebra

   $${\Sigma }_0\left(T\right)=\left\{A\in \Sigma :T^n{\mathbf{1}}_A=\mathit{\text{characteristic}}\mathit{\text{function}}\forall n\ge 0\right\}$$E53

   has at most finitely many atoms.

3. For each atom $W\in {\mathrm{\Sigma }}_0\left(T\right)$,

$$\underset{n\to \infty }{lim}\mu \left(A\backslash supp\left(T^{\mathit{dn}}{\mathbf{1}}_B\right)\right)=0,$$E54

for all subset $A,B\subset W$ with $\mu \left(A\right),\mu \left(B\right)>0$, where d is the least common multiple of orders of atoms in ${\Sigma }_0\left(T\right)$.

Then T is constrictive on $L^p\left(\Omega \right)$ and

$$\begin{array}{l}{L}^p\left(\Omega \right)=E_{\mathit{rev}}\left(T\right)\oplus E_{\mathit{fl}}\left(T\right),\\ E_{\mathit{rev}}\left(T\right)=\overline{\mathrm{spn}}\left\{{\mathbf{1}}_W:W\in {\Sigma }_0\left(T\right)\mathit{\text{is}}\mathit{\text{atom}}\right\},\mathit{\text{and}}\\ E_{\mathit{fl}}\left(T\right)=\left\{x\in L^p\left(\Omega \right):\underset{n\to \infty }{lim}\parallel T^{n}x{\parallel }_p=0\right\}.\end{array}$$E55