Ergodicities of Infinite Dimensional Nonlinear Stochastic Operators

Abstract

In the present paper, we introduce two classes ${\mathcal {L}}^+$ and ${\mathcal {L}}^-$ of nonlinear stochastic operators acting on the simplex of $\ell ^1$-space. For each operator V from these classes, we study omega limiting sets $\omega _V$ and $\omega _V^{(w)}$ with respect to $\ell ^1$-norm and pointwise convergence, respectively. As a consequence of the investigation, we establish that every operator from the introduced classes is weak ergodic. However, if V belongs to ${{\mathcal {L}}}^-$, then it is not ergodic (w.r.t $\ell ^1$-norm) while V is weak ergodic.

Appendix A. Pointwise Convergence on \(\ell ^1\)

In this section is devoted to some properties of point-wise convergence in $\ell ^1$.

It is known that $S=convh(Extr S)$, where Extr(S) is the extremal points of S and convh(A) is the convex hall of a set A. Any extremal point of S has the following form:

$$\begin{aligned} {{\mathbf {e}}}_k=(\underbrace{0,\ldots ,0,1}_{k},0,0,\ldots ), \ \ \forall k\in {\mathbb {N}}. \end{aligned}$$

Here and henceforth we denote

$$\begin{aligned} riS_r = \left\{ {{\mathbf {x}}}\in S_r : x_{k}>0,\ k\in {\mathbb {N}}\right\} , \ \ \partial S_r = S_r{\setminus } riS_r. \end{aligned}$$

Let $\{{{\mathbf {x}}}^{(n)}\}_{n\ge 1}$ be a sequence in $\ell _1$. In what follows we write ${{\mathbf {x}}}^{(n)}{\mathop {\longrightarrow }\limits ^{\left\| \cdot \right\| }}{{\mathbf {a}}}$ instead of $\left\| {{\mathbf {x}}}^{(n)}-{{\mathbf {a}}}\right\| \rightarrow 0$.

Remark A.1

Note that for any $r>0$ the sets $S_r$ and $B_r$ are not compact w.r.t. $\ell ^1$-norm. In the finite dimensional setting, analogues of these sets are compact, and hence, the investigation of the dynamics of nonlinear mappings over these kind of sets use well-known methods and techniques of dynamical systems. In our case, the non compactness (w.r.t. $\ell ^1$-norm) of the set $\mathbf{B}_r^+$ complicates our further investigation on dynamics of Volterra operators. Therefore, we need such a weak topology on $\ell ^1$ so that the set $\mathbf{B}_r^+$ would be compact with respect to that topology.

One of weak topologies on $\ell ^1$ is the Tychonov topology which generates the pointwise convergence. We say that a sequence $\{{{\mathbf {x}}}^{(n)}\}_{n\ge 1}\subset \ell ^1$ converges pointwise to ${{\mathbf {x}}}=(x_1,x_2,\ldots )\in \ell ^1$ if

$$\begin{aligned} \lim _{n\rightarrow \infty }x^{(n)}_k=x_k\ \ \ \text{ for } \text{ every } k\ge 1. \end{aligned}$$

and write ${{\mathbf {x}}}^{(n)}{\mathop {\longrightarrow }\limits ^{\mathrm {p.w.}}}{{\mathbf {x}}}$.

Remark A.2

We notice that the set $\ell ^1$ is not closed w.r.t. pointwise topology, and its completion is s which is the space of all sequences. It is known that this topology is metrizable by the following metric:

$$\begin{aligned} \rho ({{\mathbf {a}}},\mathbf{b})=\sum _{k=1}^\infty 2^{-k}\frac{|a_k-b_k|}{1+|a_k-b_k|},\ \ \ {{\mathbf {a}}},\mathbf{b}\in s. \end{aligned}$$

(A.1)

Hence, for a given sequence $\{{{\mathbf {x}}}^{(n)}\}_{n\ge 1}\subset s$ the following statements are equivalent:

• (i) ${{\mathbf {x}}}^{(n)}{\mathop {\longrightarrow }\limits ^{\mathrm {p.w.}}}{{\mathbf {x}}}$;

• (ii) ${{\mathbf {x}}}^{(n)}{\mathop {\longrightarrow }\limits ^{\rho }}{{\mathbf {x}}}$.

In the sequel, we will show that the unit ball of $\ell ^1$ is compact w.r.t. pointwise convergence, while whole $\ell ^1$ is not closed in s.

We recall that $\ell ^\infty$ is defined to be the space of all bounded sequences endowed with the norm

$$\begin{aligned} \left\| {{\mathbf {x}}}\right\| _{\infty }=\sup \left\{ |x_{n}|: n\in {\mathbb {N}}\right\} . \end{aligned}$$

The following lemma plays a crucial role in our further investigations.

Lemma A.3

Let $\{{{\mathbf {x}}}^{(n)}\}_{n\ge 1}\subset S_r$, for some $r>0$. If ${{\mathbf {x}}}^{(n)}{\mathop {\longrightarrow }\limits ^{\left\| \cdot \right\| }}{{\mathbf {a}}}$, then ${{\mathbf {a}}}\in S_r$.

Proof

It is easy to check that $\left\| {{\mathbf {x}}}-{{\mathbf {y}}}\right\| \ge |r-\rho |$, $\forall {{\mathbf {x}}}\in S_r,\ \forall {{\mathbf {y}}}\in S_\rho$. This fact together with ${{\mathbf {x}}}^{(n)}{\mathop {\longrightarrow }\limits ^{\left\| \cdot \right\| }}{{\mathbf {a}}}$ yields that ${{\mathbf {a}}}\in S_r$. $\square$

Proposition A.4

The set $\mathbf{B}_1^+$ is sequentially compact w.r.t. the pointwise convergence.

It is clear that ${{\mathbf {x}}}^{(n)}{\mathop {\longrightarrow }\limits ^{\left\| \cdot \right\| }}{{\mathbf {a}}}$ implies ${{\mathbf {x}}}^{(n)}{\mathop {\longrightarrow }\limits ^{\mathrm {p.w.}}}{{\mathbf {a}}}$. A natural question arises: is there any equivalence criteria for these two types of convergence on some set? Next result gives a positive answer to this question.

Lemma A.5

Let $\{{{\mathbf {x}}}^{(n)}\}_{n\ge 1}$ be a sequence on $S_r$. Then the following statements are equivalent:

• (1) ${{\mathbf {x}}}^{(n)}{\mathop {\longrightarrow }\limits ^{\left\| \cdot \right\| }}{{\mathbf {a}}}$ and ${{\mathbf {a}}}\in S_r$;

• (2) ${{\mathbf {x}}}^{(n)}{\mathop {\longrightarrow }\limits ^{\mathrm {p.w.}}}{{\mathbf {a}}}$ and ${{\mathbf {a}}}\in S_r$.

Recall that a functional $\varphi :\ell ^1\rightarrow {\mathbb {R}}$ is called pointwise continuous if for any ${{\mathbf {a}}}\in \ell ^1$ and any sequence $\{{{\mathbf {x}}}^{(n)}\}_{n\ge 1}\subset \ell ^1$ with ${{\mathbf {x}}}^{(n)}{\mathop {\longrightarrow }\limits ^{\mathrm {p.w.}}}{{\mathbf {a}}}$ one has $\varphi ({{\mathbf {x}}}^{(n)})\rightarrow \varphi ({{\mathbf {a}}})$.

Now we provide a criteria for linear functionals to be pointwise continuous.

Given $\mathbf{b}\in \ell ^\infty$, let us define

$$\begin{aligned} \varphi _\mathbf{b}({{\mathbf {x}}})=\sum _{k=1}^\infty b_kx_k,\ \ \ \ {{\mathbf {x}}}\in \ell ^1. \end{aligned}$$

(A.2)

Lemma A.6

Let $\mathbf{b}\in \ell ^\infty$, then the linear functional $\varphi _\mathbf{b}$ is pointwise continuous on $\mathbf{B}_1^+$ iff $\mathbf{b}\in c_0$.

Proof

Assume that $\varphi _\mathbf{b}$ is a pointwise continuous. Consider the sequence $\{{{\mathbf {e}}}_n\}_{n\ge 1}$ for which one has ${{\mathbf {e}}}_n{\mathop {\longrightarrow }\limits ^{\mathrm {p.w.}}}\mathbf {0}$, where $\mathbf{{0}}=(0,0,\ldots )$. From $\varphi _\mathbf{b}({{\mathbf {e}}}_n)=b_n$, $\varphi _\mathbf{b}(\mathbf{{0}})=0$ and the pointwise continuity of $\varphi _\mathbf{b}$ implies $b_n\rightarrow 0$ as $n\rightarrow \infty$.

Now let us suppose that $b_k\rightarrow 0$ as $k\rightarrow \infty$, and take any sequence $\{{{\mathbf {x}}}^{(n)}\}_{n\ge 1}\subset \mathbf{B}_r^+$ such that ${{\mathbf {x}}}^{(n)}{\mathop {\longrightarrow }\limits ^{\mathrm {p.w.}}}{{\mathbf {x}}}$. We will show that $\varphi _\mathbf{b}({{\mathbf {x}}}^{(n)})\rightarrow \varphi _\mathbf{b}({{\mathbf {x}}})$. If $\left\| \mathbf{b}\right\| _\infty =0$ then nothing to proof. So, we consider $\left\| \mathbf{b}\right\| _\infty \ne 0$.

Take an arbitrary positive number $\varepsilon$. Then there exists an integer $m\ge 1$ such that $|b_k|<\frac{\varepsilon }{4r}$ for all $k>m$. The pointwise convergence ${{\mathbf {x}}}^{(n)}{\mathop {\longrightarrow }\limits ^{\mathrm {p.w.}}}{{\mathbf {x}}}$ implies the existence of an integer $n_0$ such that

$$\begin{aligned} |x^{(n)}_k-x_k|<\frac{\varepsilon }{2\left\| \mathbf{b}\right\| _\infty },\ \ k\in \{1,\ldots ,m\},\ \ \ \forall n>n_0. \end{aligned}$$

Consequently, we have

$$\begin{aligned} \left| \varphi _\mathbf{b}({{\mathbf {x}}}^{(n)})-\varphi _\mathbf{b}({{\mathbf {x}}})\right|\le & {} \left| \sum _{k\le m}b_k(x_k^{(n)}-x_k)\right| + \left| \sum _{k>m}b_k(x_k^{(n)}-x_k)\right| \\\le & {} \sum _{k\le m}\left| b_k(x_k^{(n)}-x_k)\right| +\sum _{k>m}\left| b_k(x_k^{(n)}-x_k)\right| \\\le & {} \left\| \mathbf{b}\right\| _\infty \sum _{k\le m}\left| x_k^{(n)}-x_k\right| +\frac{\varepsilon }{4r}\sum _{k>m}\left| x_k^{(n)}-x_k\right| \\< & {} \left\| \mathbf{b}\right\| _\infty \cdot \frac{\varepsilon }{2\left\| \mathbf{b}\right\| }_\infty +\frac{\varepsilon }{4r}\cdot 2r\\= & {} \varepsilon ,\ \ \ \text {for all} \ \ n>n_0. \end{aligned}$$

This yields the desired assertion. $\square$