Asymptotic stability for non-equicontinuous Markov semigroups

TL;DR

This paper introduces a new criterion for the asymptotic stability of non-equicontinuous Markov semigroups, including an example with essential randomness demonstrating stability.

Contribution

It formulates a novel stability criterion for eventually continuous semigroups and provides an explicit example with essential randomness.

Findings

Established a new criterion for asymptotic stability.

Provided an example of a non-equicontinuous semigroup with essential randomness.

Demonstrated stability under the new criterion.

Abstract

We formulate a new criterion of the asymptotic stability for some non-equicontinuous Markov semigroups, the so-called eventually continuous semigroups. In particular, we provide a non-equicontinuous Markov semigroup example with essential randomness, which is asymptotically stable.

Equations

$$\underset{x\to z}{\lim }\underset{t\to \infty }{\mathrm{limsup}}|P_{t}f(x)-P_{t}f(z)|=0.$$

$$\underset{x\in \mathcal{X}}{\inf }\underset{t\to \infty }{\mathrm{liminf}}{P}_t(x,B(z,ϵ))>0.$$

$$\underset{t\to \infty }{\mathrm{limsup}}|P_{t}f(x)-P_{t}f(z)|<ϵ/2\text{for all }x\in B(z,\delta ).$$

$$\underset{t\to \infty }{\mathrm{liminf}}{P}_t\nu (B(z,\delta ))\ge {\int }_{\mathcal{X}}\underset{t\to \infty }{\mathrm{liminf}}{P}_t(y,B(z,\delta ))\nu (dy)>\alpha .$$

$$P_{t_1+\cdots +t_k}{\delta }_{x_j}(\cdot )=\alpha P_{t_2+\cdots +t_k}{\nu }_1^{x_j}(\cdot )+\alpha (1-\alpha )P_{t_3+\cdots +t_k}{\nu }_2^{x_j}(\cdot )+\cdots +\alpha (1-\alpha {)}^{k-1}{\nu }_k^{x_j}(\cdot )+(1-\alpha {)}^k{\mu }_k^{x_j}(\cdot ),$$

$$\underset{t\to \infty }{\mathrm{limsup}}|⟨P_{t}f,{\nu }_i^{x_1}⟩-⟨P_{t}f,{\nu }_i^{x_2}⟩|\le \underset{t\to \infty }{\mathrm{limsup}}|⟨P_{t}f-P_{t}f(z),{\nu }_i^{x_1}⟩|+|⟨P_{t}f-P_{t}f(z),{\nu }_i^{x_2}⟩|\le ϵ.$$

$$3ϵ$$

$$=\underset{t\to \infty }{\mathrm{limsup}}|⟨P_{t}f,P_{t_1+\cdots +t_k}{\delta }_{x_1}⟩-⟨P_{t}f,P_{t_1+\cdots +t_k}{\delta }_{x_2}⟩|$$

$$\le \alpha \underset{t\to \infty }{\mathrm{limsup}}|⟨P_{t}f,{\nu }_1^{x_1}⟩-⟨P_{t}f,{\nu }_1^{x_2}⟩|+\cdots +\alpha (1-\alpha {)}^k\underset{t\to \infty }{\mathrm{limsup}}|⟨P_{t}f,{\nu }_k^{x_1}⟩-⟨P_{t}f,{\nu }_k^{x_2}⟩|+ϵ$$

$$\le (\alpha +\cdots +\alpha (1-\alpha {)}^k)ϵ+ϵ<2ϵ,$$

$$f_i(y):=\rho (y,(K_i^{ϵ/4}{)}^c)/(\rho (y,(K_i^{ϵ/4}{)}^c)+\rho (y,K_i)),$$

$$1_{K_i}\le f_i\le 1_{K_i^{ϵ/4}}\text{and}\text{Lip}(f_i)\le 4/ϵ\text{for any}i\in \mathbb{N}.$$

$$\underset{k\to \infty }{\mathrm{limsup}}{P}_{t_{j_k}}f(y_0)\le ϵ/2.$$

$$P_{t_{j_k}}f(z)\ge P_{t_{j_k}}{f}_{j_k}(z)\ge P_{t_{j_k}}(z,K_{j_k})\ge ϵ>ϵ/2\text{for all}k\ge 0,$$

$$ϵ{\alpha }_0/16\ge \underset{i\to \infty }{\mathrm{liminf}}{P}_{t_i}(z,K_{j_0}^{ϵ/4})=\underset{i\to \infty }{\mathrm{liminf}}{\int }_{\mathcal{X}}{P}_{t_{j_0}}(y,K_{j_0}^{ϵ/4})P_{t_i}(z,dy)\ge \underset{i\to \infty }{\mathrm{liminf}}{\int }_{B_0}{P}_{t_{j_0}}{f}_{j_0}(y)P_{t_i}(z,dy).$$

$$ϵ{\alpha }_0/16\ge \underset{i\to \infty }{\mathrm{liminf}}{\int }_{B_0-A_0}{P}_{t_{j_0}}{f}_{j_0}(y)P_{t_i}(z,dy)\ge \underset{i\to \infty }{\mathrm{liminf}}{P}_{t_i}(z,B_0-A_0)\cdot ϵ/8,$$

$${\alpha }_1:=\underset{t\to \infty }{\mathrm{liminf}}{P}_t(z,B_1)\ge \underset{t\to \infty }{\mathrm{liminf}}{\int }_{B(z,\delta )}{P}_s(z^{\prime },B_1)P_t(z,d{z}^{\prime })\ge \frac{1}{2}{P}_s(z,B_1)\cdot \underset{t\to \infty }{\mathrm{liminf}}{P}_t(z,B(z,\delta ))>0.$$

$$P_{t_{j_k}}{f}_{j_k}(y_0)\le ϵ/8,\forall k\ge 0.$$

$$P_{t_{j_k}}\sum _{l\ge k}{f}_{j_l}(y_0)\le ϵ/4,\forall k\ge 0.$$

$$\underset{k\to \infty }{\mathrm{limsup}}{P}_{t_{j_k}}{f}_{j_l}(y_0)\le ϵ/(8\cdot 2^l),\forall l\ge 0.$$

$$P_{t_{j_k}}\sum _{0\le l<k}{f}_{j_l}(y_0)\le \sum _{0⩽l<k}{2}^{-l}ϵ/8\le ϵ/4.$$

$$P_{t_{j_k}}\sum _{l\ge 0}{f}_{j_l}(y_0)\le ϵ/2,\forall k\ge 0,$$

$$p_{n^{-1},n^{-1}}(t)=p_{n,n}(t)=e^{-t/n},$$

$$p_{n^{-1},0}(t)=1-p_{n^{-1},n^{-1}}(t)-p_{n^{-1},n}(t),$$

$$P_{n}f(n^{-1})-P_{n}f(0)\ge e^{-1},\forall n\ge 2.$$

$$(p_1(x),p_2(x),p_3(x))=\{\begin{array}{ll}(\frac{x}{2},1-x,\frac{x}{2}),& 0\le x<\frac{2}{3},\\ (\frac{1}{3},\frac{1}{3},\frac{1}{3}),& \frac{2}{3}\le x\le \frac{3}{2},\\ (\frac{1}{2x},1-x^{-1},\frac{1}{2x}),& x>\frac{3}{2}.\end{array}$$

$$P_{n}f(x_n)\ge \mathbb{P}({\Phi }_n^{x_n}=n)f(n)\ge \frac{1}{2}\lambda e^{-\lambda }>0=P_{n}f(0),$$

$$\sum _{i\in I}{p}_i(x)\rho (w_i(x),z)\le r(x)\rho (x,z)\text{ for }x\in \mathcal{X}.$$

Full text

††footnotetext: Fuzhou. Gong and Yuan Liu are supported by National Key R & D Program of China (No. 2020YFA0712700) and Key Laboratory of Random Complex Structures and Data Sciences, Mathematics and Systems Science, Chinese Academy of Sciences (No. 2008DP173182); Yong Liu is supported by CNNSF (No. 11731009, No. 12231002) and Center for Statistical Science, PKU; Yuan Liu is supported by (CNNSF 11688101).

1 Introduction

In this note, we are mainly concerned with the asymptotic stability of Markov-Feller semigroups in continuous time. Specifically, we adopt the notion of asymptotic equicontinuity condition, introduced by Jaroszewska[5], or the eventual continuity, by Gong and Liu[3], to work as a necessary condition for the asymptotic stability. In fact, these two notions are formulated almost simultaneously, and mathematically equivalent. We adopt the name of eventual continuity for simplicity, which depicts the feature that a uniquely ergodic semigroup may behave sensitively in initial data, and is evidently weaker than the e-property, i.e. equicontinuity. The eventual continuity is a reasonable and essential tool to deal with non-equicontinuous semigroups.

In 2006, Lasota and Szarek developed the lower bound technique for the e-processes (i.e. Markov processes with the e-property), to formulate a criterion for the existence of an invariant measure[7, 8]. Since then, the e-property has turned out to play a valuable role in studying ergodicity. However, since the e-property requires some uniform estimates in a neighborhood of each point, it does not hold true for general cases, or even if it does, it may be cumbersome to verify, see discussion in [2, 6]. Instead, we provide a new criterion of asymptotic stability by virtue of the eventual continuity (see Theorem 3.1 below), which extends [9, Corollary 5.4] from the e-processes to eventually continuous semigroups. In particular, we provide nontrivial examples with randomness, which are not the e-processes.

2 Preliminaries

Let $(\mathcal{X},\rho)$ be a Polish space. By $\mathcal{B}(\mathcal{X})$ we denote the $\sigma$-field of all its Borel subsets. By $B(x,r)$ we denote the open ball in $(\mathcal{X},\rho)$ of radius $r$, centred at $x\in\mathcal{X}$. By $B_{b}(\mathcal{X})$, we denote the vector space of all bounded real-valued Borel measurable functions on $\mathcal{X}$, and by $C_{b}(\mathcal{X})$ all bounded real-valued continuous functions, both equipped with the supremum norm $|\cdot|_{\infty}$. By $L_{b}(\mathcal{X})$ we denote the subspace of $C_{b}(\mathcal{X})$ of all bounded Lipschitz functions. By $\mathcal{M}_{1}(\mathcal{X})$, we denote the family of all probability measures on $\mathcal{X}$. For $\mu\in\mathcal{M}_{1}(\mathcal{X})$, its support is the set $\text{supp }\mu=\{x\in\mathcal{X}:\mu(B(x,\epsilon))>0\text{ for every }\epsilon>0\}$. Moreover, for $f\in B_{b}(\mathcal{X})$ and $\mu\in\mathcal{M}_{1}(\mathcal{X})$, we use the notation $\langle f,\mu\rangle=\int_{\mathcal{X}}f(x)\mu(dx).$

Let $T$ be the index set, $\mathbb{R}_{+}$ or $\mathbb{N}_{+}$. $\{P_{t}\}_{t\in T}$ is a Markov-Feller semigroup on $\mathcal{X},$ if $\{P_{t}\}_{t\in T}$ is a Markov semigroup and $P_{t}(C_{b}(\mathcal{X}))\subset C_{b}(\mathcal{X})$ for $t\in T$. Let $\{P^{*}_{t}\}_{t\in T}$ be the dual semigroup defined on $\mathcal{M}_{1}(\mathcal{X})$ by the formula $P^{*}_{t}\mu(B):=\int_{\mathcal{X}}P_{t}\mathbf{1}_{B}d\mu$ for $B\in\mathcal{B}(\mathcal{X}).$ For ease of notation, we simply rewrite $P^{*}_{t}$ to $P_{t}$. Recall that $\mu\in\mathcal{M}_{1}(\mathcal{X})$ is invariant for the semigroup $\{P_{t}\}_{t\in T}$ if $P_{t}\mu=\mu$ for all $t\in T$. $\{P_{t}\}_{t\in T}$ is asymptotically stable if there exists a unique invariant measure $\mu\in\mathcal{M}_{1}(\mathcal{X})$, and $P_{t}\nu$ converges weakly to $\mu$ as $t\rightarrow\infty$ for every $\nu\in\mathcal{M}_{1}(\mathcal{X})$.

$\{P_{t}\}_{t\in T}$ is said to be eventually continuous at $z\in\mathcal{X},$ if for an arbitrary $f\in L_{b}(\mathcal{X})$,

[TABLE]

3 Main results

Throughout this paper, we assume that $\{P_{t}\}_{t\in T}$ is a Markov-Feller semigroup. One interesting observation gives that the eventual continuity is necessary for the asymptotic stability, which can be proved by merely comparing the definitions between these two notions. Moreover, for eventually continuous semigroups, the asymptotic stability can be derived from some lower bound condition on probability transition. Our principal result is a new criterion of asymptotic stability for the eventual continuous semigroups as follows.

Theorem 3.1**.**

The following three statements are equivalent:

• $(i)$

$\{P_{t}\}_{t\in T}$* is asymptotically stable with unique invariant measure $\mu$.*

• $(ii)$

$\{P_{t}\}_{t\in T}$* is eventually continuous on $\mathcal{X}$. Moreover, for all $z\in{\rm supp}\;\mu$, and all $\epsilon>0$,*

[TABLE]

• $(iii)$

$\{P_{t}\}_{t\in T}$* is eventually continuous on $\mathcal{X}$. Moreover, there exists some $z\in\mathcal{X}$, such that for all $\epsilon>0$, (1) holds.*

Proof.

The implication that $(i)\Rightarrow(ii)$ follows from [9, Corollary 5.4]. $(ii)\Rightarrow(iii)$ is trivial. It remains to show that $(iii)\Rightarrow(i)$, which consists of two parts respectively about the asymptotic behavior of semigroups and the existence of invariant measure.

Part 1. First we show that $\lim\limits_{t\rightarrow\infty}|P_{t}f(x)-P_{t}f(y)|=0$ for all $x,y\in\mathcal{X}$ and all $f\in L_{b}(\mathcal{X}).$ Assume, contrary to our claim, there exists some $x_{1},x_{2}\in\mathcal{X}$, $f\in L_{b}(\mathcal{X})$ and $\epsilon>0$ such that $\limsup\limits_{t\rightarrow\infty}|P_{t}f(x_{1})-P_{t}f(x_{2})|\geq 3\epsilon$. As $\{P_{t}\}_{t\in T}$ is eventually continuous at $z$, there is $\delta>0$ such that

[TABLE]

For such $\delta>0$, Condition (1) gives some $\alpha\in(0,\frac{1}{2})$ such that $\liminf\limits_{t\rightarrow\infty}P_{t}(x,B(z,\delta))>\alpha$ for all $x\in\mathcal{X}$. Then Fatou’s lemma gives, for any $\nu\in\mathcal{M}_{1}(\mathcal{X})$ with $\text{supp }\nu\subset B(z,\delta),$

[TABLE]

Let $k\geq 1$ be such that $2(1-\alpha)^{k}|f|_{\infty}<\epsilon.$ By induction we are going to define four sequences of measures $\{\nu_{i}^{x_{1}}\}_{i=1}^{k},\{\mu_{i}^{x_{1}}\}_{i=1}^{k},\{\nu_{i}^{x_{2}}\}_{i=1}^{k},\{\mu_{i}^{x_{2}}\}_{i=1}^{k},$ and a sequence of positive numbers $\{t_{i}\}_{i=1}^{k}$ in the following way: let $t_{1}>0$ be such that $P_{t}(x_{j},B(z,\delta))>\alpha$ for all $t\geq t_{1}$, $j=1,2$. Set

$\nu_{1}^{x_{j}}(\cdot)=\dfrac{P_{t_{1}}\delta_{x_{j}}(\cdot\cap B(z,\delta))}{P_{t_{1}}\delta_{x_{j}}(B(z,\delta))},\quad$ $\mu_{1}^{x_{j}}(\cdot)=\dfrac{1}{1-\alpha}(P_{t_{1}}\delta_{x_{j}}(\cdot)-\alpha\nu_{1}^{x_{j}}(\cdot)),\quad j=1,2.$

Assume that we have done it for $i=1,\dots,l,$ for some $l<k.$ Now let $t_{l+1}\geq t_{l}$ be such that $P_{t}\mu_{l}^{x_{j}}(B(z,\delta))>\alpha$ for all $t\geq t_{l+1}$, $j=1,2$. Set

$\nu_{l+1}^{x_{j}}(\cdot)=\dfrac{P_{t_{l+1}}\mu_{l}^{x_{j}}(\cdot\cap B(z,\delta))}{P_{t_{l+1}}\mu_{l}^{x_{j}}(B(z,\delta))},\quad$ $\mu_{l+1}^{x_{j}}(\cdot)=\dfrac{1}{1-\alpha}(P_{t_{l+1}}\mu_{l}^{x_{j}}(\cdot)-\alpha\nu_{l+1}^{x_{j}}(\cdot)),\quad j=1,2.$

Then it follows that

[TABLE]

where $\text{supp }{{\nu}_{i}^{x_{j}}}\subset B(z,\delta),\,i=1,\dots,k,\,j=1,2.$ Thus from the Fatou’s lemma and eventual continuity, we have

[TABLE]

Finally, using the measure decomposition gives

[TABLE]

which is impossible.

Part 2. To show the existence of invariant measure, it suffices to show that $\{P_{t}(z,\cdot)\}_{t\in T}$ is tight. Assume, on the contrary, $\{P_{t}(z,\cdot)\}_{t\in T}$ is not tight, then by the arguments given in [7], there exists some $\epsilon>0,$ a sequence of compact sets $\{K_{i}:i\geq 1\}$ and an increasing sequence of real numbers $\{t_{i}:i\geq 1\},t_{i}\rightarrow\infty,$ such that $P_{t_{i}}(z,K_{i})\geq\epsilon$, for all $i\geq 1$ and $K_{i}^{\epsilon/2}\cap K_{j}^{\epsilon/2}=\emptyset$ for any $i\neq j$, where $K_{i}^{\epsilon/2}:=\{x\in\mathcal{X}:\inf_{y\in K_{i}}\rho(x,y)<\epsilon/2\}$. Define $f_{i}\in L_{b}(\mathcal{X})$ by

[TABLE]

satisfying that

[TABLE]

We claim that there exists a subsequence of integers $\{j_{k}:k\geq 0\}$ and some point $y_{0}\in\mathcal{X}$ such that for $f:=\sum_{l\geq 0}f_{j_{l}}\in L_{b}(\mathcal{X})$

[TABLE]

Admitting this claim (3) for the moment, it follows from the definition of $f_{j_{k}}$ that

[TABLE]

which contradicts the asymptotic result in Part 1. This completes the proof of tightness.

Following some arguments in [3, Theorem 1.6], we prove the claim (3) as follows.

$\mathbf{Step\;1}.$ Denote $B_{0}=B(z,r).$ By (1), let $\alpha_{0}:=\liminf\limits_{i\rightarrow\infty}P_{t_{i}}(z,B_{0})>0$. Lemma 2.2 in [3] yields that there exists $j_{0}$ sufficiently large such that

[TABLE]

Due to the Feller property, define an open subset $A_{0}=\{y\in B_{0}:P_{t_{j_{0}}}f_{j_{0}}(y)<\epsilon/8\}$. It follows that

[TABLE]

which implies $\liminf\limits_{i\rightarrow\infty}P_{t_{i}}(z,B_{0}-A_{0})\leq\alpha_{0}/2$, and thus $\limsup\limits_{i\rightarrow\infty}P_{t_{i}}(z,A_{0})\geq\alpha_{0}/2$. Hence $A_{0}$ is nonempty, which contains a ball $B_{1}$ of radius less than $r/2$ such that $P_{t_{j_{0}}}f_{j_{0}}(y)\leq\epsilon/8$ for all $y\in\overline{B_{1}}$ and $P_{s}(z,B_{1})>0$ for some $s>0$. By the Feller property, there exists $\delta>0$ such that for any $z^{\prime}\in B(z,\delta),$ $P_{s}(z^{\prime},B_{1})\geq\frac{1}{2}P_{s}(z,B_{1})>0$. From Condition (1), we derive that

[TABLE]

Inductively, for each $k\geq 0,$ we can determine $B_{k+1}\subset B_{k}$ of radius less than $r/2^{k+1}$ and increasing $j_{k}\in\mathbb{N}$ such that $P_{t_{j_{k}}}f_{j_{k}}(y)\leq\epsilon/8$ for all $y\in\overline{B_{k+1}}$. Therefore, we can find a common $y_{0}\in\bigcap_{k\geq 0}\overline{B_{k}}$ such that

[TABLE]

$\mathbf{Step\;2}.$ Noting that all $K_{i}^{\epsilon/2}$ are disjoint mutually, there exists some $u_{0}$ sufficiently large such that $P_{t_{j_{0}}}\sum_{l\geq u_{0}}f_{j_{l}}(y_{0})\leq\epsilon/8$. Combining with (4) yields $P_{t_{j_{0}}}(f_{t_{j_{0}}}+\sum_{l\geq u_{0}}f({j_{l}}))(y_{0})\leq\epsilon/4$. Similarly, there exists $u_{1}>u_{0}$ such that $P_{t_{j_{u_{0}}}}(f_{t_{j_{u_{0}}}}+\sum_{l\geq u_{1}}f({j_{l}}))(y_{0})\leq\epsilon/4$. By induction, we obtain a subsequence $\{j_{0},j_{u_{0}},j_{u_{1}},\ldots\}\subset\{j_{l}\}$, still denoted by $\{j_{l}\}$ for simplicity, such that

[TABLE]

On the other hand, by [3, Lemma 2.3] and (4), applying diagonal arguments, passing to a subsequence if necessary, we further obtain

[TABLE]

By (6), there exists some $v_{0}$ sufficiently large such that $P_{t_{j_{k}}}f_{j_{0}}(y_{0})\leq\epsilon/8$ for all $k\geq v_{0}$. Similarly, there exists $v_{1}>v_{0}$ such that $P_{t_{j_{k}}}(f_{j_{0}}+f_{j_{v_{0}}})(y_{0})\leq(1+2^{-1})\epsilon/8$ for all $k\geq v_{1}$. By induction, we derive a subsequence $\{j_{0},j_{v_{0}},j_{v_{1}},\ldots\}\subset\{j_{l}\}$, still denoted by $\{j_{l}\}$, satisfying that

[TABLE]

Combining (7) with (5), it yields

[TABLE]

which completes the proof of our claim (3). ∎

Example 3.2**.**

We adopt the idea in [4, Example 2.1] to construct a non-equicontinuous Markov chain in continuous-time. Let $\mathcal{X}=\{0\}\cup\{n^{-1}:n\geq 2\}\cup\{n:n\geq 2\}.$ Define a continuous-time Markov chain $\{\Phi_{t}\}_{t\geq 0}$ with transition rate matrix $\{q_{ij}\}_{i,j\in\mathcal{X}},$ where $q_{n,n}=q_{n^{-1},n^{-1}}=-n,\;q_{n,0}=q_{n^{-1},n}=n$, for $n\geq 2,$ and $q_{i,j}=0$ otherwise. The transition probability $\{p_{ij}(t)\}_{i,j\in\mathcal{X}}$ is defined as follows: $p_{0,0}(t)=1$; for $n\geq 2$,

[TABLE]

It can be checked that $\{P_{t}\}_{t\geq 0}$ generated by $\{\Phi_{t}\}_{t\geq 0}$ is a Markov-Feller semigroup. Further, since $\Phi_{t}$ converges to [math] almost surely as $t\rightarrow\infty$, $\{P_{t}\}_{t\geq 0}$ is asymptotically stable, hence $\{P_{t}\}_{t\geq 0}$ is eventually continuous on $\mathcal{X}$ by Theorem 3.1. However, the e-property fails at [math], since we can take $f(x)=x\land 1\in C_{b}(\mathcal{X})$ and then

[TABLE]

4 Application to iterated function systems (IFS) with jumps

We apply Theorem 3.1 to investigate the asymptotic stability of non-equicontinuous Markov semigroups. First of all, we construct such kind of semigroups with essential randomness (see Example 4.1). The randomness comes from that, at each step, the dynamical system jumps to different positions, obeying some place-dependent probabilistic distribution. Furthermore, developing these ideas, we extend it to a class of place-dependent iterated function systems (IFS). Specifically, we modify the model introduced in [1], to a general IFS with a place-dependent contraction coefficient, and apply Theorem 3.1 to prove the asymptotic stability.

We first introduce some notations. Let $(\mathcal{X},\rho)$ be a Polish space, $I=\{1,\dots,N\}$ be a finite set, $w_{i}:\mathcal{X}\rightarrow\mathcal{X},\;i\in I$ be transformations on $\mathcal{X}$, and $\{S(t)\}_{t\geq 0}$ be a continuous flow on $\mathcal{X}$. Let $(p_{1}(x),\dots,p_{N}(x))$ be a probabilistic vector denoted by $p(x)$, that is, for each $x\in\mathcal{X}$, $p_{i}(x)\geq 0$ for $i\in I$ and $\sum_{i\in I}p_{i}(x)=1$. Let $\{\tau_{n}\}_{n\geq 1}$ be a sequence of random variables with $\tau_{0}=0,$ and $\triangle\tau_{n}=\tau_{n}-\tau_{n-1},n\geq 1$, are i.i.d. with density $\lambda e^{-\lambda t},\;\lambda>0$.

Now we define the $\mathcal{X}$-valued Markov chain $\Phi=\{\Phi^{x}(t):x\in\mathcal{X}\}_{t\geq 0}$ in the following way.

• $\mathbf{Step\;1.}$

Let $x\in\mathcal{X}$. Denote $\Phi_{0}^{x}:=x$.

• $\mathbf{Step\;2.}$

Let $\xi_{1}=S(\tau_{1})(x).$ We have a random index $i_{1}\in I$ with probability $p(\xi_{1})$, i.e. $\mathbb{P}(i_{1}=k)=p_{k}(\xi_{1})$, for $k\in I$. Set $\Phi_{1}^{x}:=w_{i_{1}}(\xi_{1}).$

• $\mathbf{Step\;3.}$

Recursively, assume that $\Phi_{1}^{x},\dots,\Phi_{n-1}^{x},n\geq 2$ are given. We define $\xi_{n}=S(\triangle\tau_{n})(\Phi_{n-1}^{x}).$ Further, we randomly choose $i_{n}\in I$ with probability $p(\xi_{n})$, and let $\Phi_{n}^{x}=w_{i_{n}}(\xi_{n}).$

• $\mathbf{Step\;4.}$

Finally, set $\Phi^{x}(t):=S(t-\tau_{n})(\Phi_{n}^{x})\quad\text{for }\tau_{n}\leq t<\tau_{n+1},\;n\geq 0.$

Now we can define semigroup $\{P_{t}\}_{t\geq 0}$ by $P_{t}f(x):=\mathbb{E}f(\Phi^{x}(t))\text{ for }f\in B_{b}(\mathcal{X})$.

Given the above setting, we provide the following example with essential randomness which is not an e-process, but is eventually continuous.

Example 4.1**.**

Let $\mathcal{X}=\mathbb{R}_{+}.\;$ Let $S(t)=\text{Id}_{\mathcal{X}}$ for $t\geq 0,$ and $w_{1}(x)=0,$ $w_{2}(x)=x,$ $w_{3}(x)=x^{-1}\mathbf{1}_{\{x\neq 0\}}$. Let $p_{i}\in C(\mathcal{X},[0,1]),$ $i=1,2,3$, be such that

[TABLE]

Hence, any trajectory starting from $x\neq 0$ runs over $\{0,x,x^{-1}\}$. Noting that [math] is the unique absorbing state for this process, $\{P_{t}\}_{t\geq 0}$ is asymptotically stable, and thus eventually continuous on $\mathcal{X}$. However, the e-property fails at $0.$For illustration, one may take $x_{n}=1/n,$ $f(x)=x\wedge 1\in C_{b}(\mathcal{X}),$ to see that for $t=n$

[TABLE]

which conflicts with the e-property.

As this example have showed, generally, this place-dependent IFS may not be an e-process. In fact, under some wild assumptions, it is straightforward to establish the eventual continuity. We first introduce some notations. For $\mathbf{i}=(i_{1},\dots,i_{n})\in I^{n},\,n\geq 1$, denote $\mathbf{i}[k]=(i_{1},\dots,i_{k})$ for $0\leq k\leq n$, $w_{\mathbf{i}[0]}(x)=x$, $w_{\mathbf{i}[k]}(x)=w_{i_{1}}\circ\cdots\circ w_{i_{k}}(x)$ for $1\leq k\leq n$, $w_{\mathbf{i}}(x)=w_{\mathbf{i}[n]}(x)$ and $J_{n}(x):=\max\limits_{\mathbf{i}\in I^{n}}\Pi_{j=0}^{n-1}r(w_{\mathbf{i}[j]}(x))$. Now we can define an IFS with a place-dependent contraction coefficient. Our assumptions are listed below.

• $\mathbf{B_{1}.}$

$w_{i},\,p_{i}$ are continuous, for $i\in I$.

• $\mathbf{B_{2}.}$

There exists $z\in\mathcal{X}$ and a continuous function $r:\;\mathcal{X}\rightarrow(0,1]$ such that

[TABLE]

• $\mathbf{B_{3}.}$

There exists a function $\omega:\;\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ such that $\sum_{i\in I}|p_{i}(x)-p_{i}(z)|\leq\omega(\rho(x,z))$ for $x\in\mathcal{X}$, where $\omega$ is concave, non-decreasing and $\omega(0)=0$.

• $\mathbf{B_{4}.}$

There exists $\alpha\geq 0$ such that $\rho(S(t)(x),S(t)(y))\leq e^{\alpha t}\rho(x,y)$, for$x,y\in\mathcal{X},\;t\geq 0$.

• $\mathbf{B_{5}.}$

There exists $M\in\mathbb{N}$ and $\eta>0$ such that $\sum_{n=M}^{\infty}\omega\left(J_{n}(x)\rho(x,z)\frac{\lambda^{n}}{(\lambda-\alpha)^{n}}\right)\leq 1-\gamma$ for $x\in B(z,\eta)$.

Here we emphasize that the contraction coefficient $r$ is depending on places and is not assumed to be strictly less than $1$. In other words, although the system may still be contracting, the pace of convergence may vary form place to place, which allows us to derive the eventual continuity rather than the e-property in a direct way.

Proposition 4.2**.**

Assume that $\mathbf{B_{1}}-\mathbf{B_{5}}$ hold, then $\{P_{t}\}_{t\geq 0}$ is eventually continuous at $z.$

It can be checked that the Markov semigroup generated by $\{\Phi_{t}^{x}\}$ is Feller. While $\mathbf{B_{1}}-\mathbf{B_{5}}$ are not enough to ensure the lower bound condition (1), we further have the following assumptions.

• $\mathbf{C_{1}.}$

$S(t)(z)=z$ for $t\geq 0.$

• $\mathbf{C_{2}.}$

For each $\epsilon>0,$ there exists $\beta=\beta(\epsilon)>0,$ and that for each $x\in\mathcal{X}$ there exists some $t_{x}=t(x,\epsilon)\geq 0$ such that $P_{t_{x}}(x,B(z,\epsilon))>\beta$.

Theorem 4.3**.**

Assume that $\mathbf{B_{1}}-\mathbf{B_{5}}$ and $\mathbf{C_{1}},\mathbf{C_{2}}$ hold, then $\{P_{t}\}_{t\geq 0}$ is asymptotically stable.

Finally we provide a simple example, which can be showed to be asymptotically stable by Theorem 4.3 immediately, since Example 4.1 does not satisfy Assumption $\mathbf{B_{1}}$.

Example 4.4**.**

Let $\mathcal{X}=\mathbb{R}_{+}.\;$ Let $S(t)=\text{Id}_{\mathcal{X}}\;$ for $t\geq 0$, and $w_{1}(x)=x/2$, $w_{2}(x)=x$. Let $p_{1}(x)=e^{-x}$ and $p_{2}(x)=1-e^{-x}$ for $x\geq 0.$ Then one may check that for $z=0$, $r(x)=1-e^{-x}/2$, $J_{n}(x)=(1-e^{-x}/2)^{n}$, $M=0$, $\eta=1/8$, $\gamma=1-e^{1/8}/4$, $\beta=\Pi_{i=1}^{\infty}(1-2^{-i})$, all the assumptions $\mathbf{B_{1}}-\mathbf{B_{5}}$ and $\mathbf{C_{1}}$, $\mathbf{C_{2}}$ are satisfied.