# Discrete-Time Semi-Markov Random Evolutions in Asymptotic Reduced Random Media with Applications

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## Abstract

**:**

## 1. Introduction

## 2. Semi-Markov Chains with Merging

## 3. Merging of Semi-Markov Chains

#### 3.1. The Ergodic Case

- C1:
- The transition kernel ${P}^{\epsilon}(x,B)$ of the embedded Markov chain ${x}_{n}^{\epsilon}$ has the representation (7).
- C2:
- The supporting Markov chain $({x}_{n}^{0})$ with transition kernel P is uniformly ergodic in each class ${E}_{j}$, with stationary distribution ${\rho}_{j}(dx)$, $j\in \widehat{E}$, that is,$${\rho}_{j}(B)={{\displaystyle \int}}_{{E}_{j}}{\rho}_{j}(dx)P(x,B),\phantom{\rule{4pt}{0ex}}\mathrm{and}\phantom{\rule{4pt}{0ex}}{\rho}_{j}({E}_{j})=1,\phantom{\rule{4pt}{0ex}}B\in {\mathcal{E}}_{j}$$
- C3:
- The average exit probabilities of the initial embedded Markov chain $({x}_{n}^{\epsilon}$) are positive, that is,$${\widehat{p}}_{j}:={{\displaystyle \int}}_{{E}_{j}}{\rho}_{j}(dx){P}_{1}(x,E\backslash {E}_{j})>0.$$
- C4:
- The mean merged values are positive and bounded, that is,$$0<{m}_{j}:={{\displaystyle \int}}_{{E}_{j}}{\rho}_{j}(dx)m(x)<\infty .$$

**Theorem**

**1.**

#### 3.2. The Non-Ergodic Case

- C5:
- The average transition probabilities of the initial embedded Markov chain $({x}_{n}^{\epsilon}$) to state 0, satisfy the following,$${\widehat{p}}_{j0}:=-{{\displaystyle \int}}_{{E}_{j}}{\rho}_{j}(dx){P}_{1}(x,E)>0$$

**Theorem**

**2.**

## 4. Semi-Markov Random Evolution

**Theorem**

**3.**

## 5. Average and Diffusion Approximation with Merging

#### 5.1. Averaging

- A1:
- The MC $({z}_{k},{\gamma}_{k},k\in \mathrm{N})$ is uniformly ergodic in each class ${E}_{j}$, with ergodic distribution ${\pi}_{j}^{\u266f}(B\times \left\{k\right\}),B\in \mathcal{E}\cap {E}_{j},k\in \mathrm{N}$, and the projector operator $\mathsf{\Pi}$ is defined by relation (10).
- A2:
- A3:
- Let us assume that the perturbed operator ${D}^{\epsilon}(x)$ has the following representation in B$$\begin{array}{c}\hfill {D}^{\epsilon}(x)=I+\epsilon {D}_{1}(x)+\epsilon {D}_{0}^{\epsilon}(x),\end{array}$$
- A4:
- We have: ${\int}_{E}\pi (dx){\u2225{D}_{1}(x)\phi \u2225}^{2}<\infty $.
- A5:
- There exists Hilbert spaces H and ${H}^{\ast}$ such that compactly embedded in Banach spaces B and ${B}^{\ast},$ respectively, where ${B}^{\ast}$ is a dual space to $B.$
- A6:
- Operators ${D}^{\epsilon}(z)$ and ${({D}^{\epsilon})}^{\ast}(z)$ are contractive on Hilbert spaces H and ${H}^{\ast},$ respectively.We note that if $B={C}_{0}(\mathrm{R})$, the space of continuous function on $\mathrm{R}$ vanishing at infinity, then $H={W}^{l,2}(\mathrm{R})$ is a Sobolev space, and ${W}^{l,2}(\mathrm{R})\subset {C}_{0}(\mathrm{R})$ and this embedding is compact (see References [34,43]). For the spaces $B={L}_{2}(\mathrm{R})$ and $H={W}^{l,2}(\mathrm{R})$ the situation is the same.

**Theorem**

**4.**

**Corollary**

**1.**

#### 5.2. Diffusion Approximation

- D1:
- Let us assume that the perturbed operators ${D}^{\epsilon}(x)$ have the following representation in B$$\begin{array}{c}\hfill {D}^{\epsilon}(x)=I+\epsilon {D}_{1}(x)+{\epsilon}^{2}{D}_{2}(x)+{\epsilon}^{2}{D}_{0}^{\epsilon}(x),\end{array}$$
- D2:
- The following balance condition holds$$\begin{array}{c}\hfill \mathsf{\Pi}{D}_{1}(x)\mathsf{\Pi}=0.\end{array}$$
- D3:

**Theorem**

**5.**

#### 5.3. Normal Deviation with Merging

**Theorem**

**6.**

## 6. Application to Particular Systems

#### 6.1. Integral Functionals

#### 6.1.1. Average Approximation

**Theorem**

**7.**

#### 6.1.2. Diffusion Approximation

**Theorem**

**8.**

#### 6.1.3. Normal Deviation

**Theorem**

**9.**

#### 6.2. Discrete Dynamical Systems

#### 6.2.1. Average Approximation

**Theorem**

**10.**

#### 6.2.2. Diffusion Approximation

**Theorem**

**11.**

#### 6.2.3. Normal Deviation

**Theorem**

**12.**

## 7. Proofs

#### 7.1. Proof of Theorem 1

**Proof.**

#### 7.2. Proof of Theorem 3

**Proof.**

#### 7.3. Proof of Theorem 4

**Proof.**

#### 7.4. Proof of Theorem 5

**Proof.**

## 8. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Limnios, N.; Swishchuk, A.
Discrete-Time Semi-Markov Random Evolutions in Asymptotic Reduced Random Media with Applications. *Mathematics* **2020**, *8*, 963.
https://doi.org/10.3390/math8060963

**AMA Style**

Limnios N, Swishchuk A.
Discrete-Time Semi-Markov Random Evolutions in Asymptotic Reduced Random Media with Applications. *Mathematics*. 2020; 8(6):963.
https://doi.org/10.3390/math8060963

**Chicago/Turabian Style**

Limnios, Nikolaos, and Anatoliy Swishchuk.
2020. "Discrete-Time Semi-Markov Random Evolutions in Asymptotic Reduced Random Media with Applications" *Mathematics* 8, no. 6: 963.
https://doi.org/10.3390/math8060963