# Poincaré-Type Inequalities for Compact Degenerate Pure Jump Markov Processes

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. The Neuroscience Framework

#### 1.2. The Model

#### 1.3. Poincaré-Type Inequalities

**Theorem**

**1.**

**Corollary**

**1.**

**Theorem**

**2.**

**Corollary**

**2.**

## 2. Proof of the Poincaré for General Initial Configurations

#### 2.1. Technical Results

**Lemma**

**1.**

- For all $t>{t}_{0},$ we have$$\sum _{y\in {D}_{x}}\frac{{\pi}_{t}^{2}({\Delta}^{i}\left(x\right),y)}{{\pi}_{t}(x,y)}\le {C}_{1}.$$
- For all $t\le {t}_{0},$ we have$$\sum _{y\in {D}_{x}\backslash \left\{{\Delta}^{i}\left(x\right)\right\}}\frac{{\pi}_{t}^{2}({\Delta}^{i}\left(x\right),y)}{{\pi}_{t}(x,y)}\le {C}_{1}$$$$\frac{{\pi}_{t}^{2}({\Delta}^{i}\left(x\right),{\Delta}^{i}\left(x\right))}{{\pi}_{t}(x,{\Delta}^{i}\left(x\right))}\le \frac{{\pi}_{t}({\Delta}^{i}\left(x\right),{\Delta}^{i}\left(x\right))}{{\pi}_{t}(x,{\Delta}^{i}\left(x\right))}\le \frac{{C}_{2}}{t}.$$

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

#### 2.2. Proof of Theorem 1

## 3. Proof of the Poincaré Inequalities for Starting Configuration on the Domain of the Invariant Measure

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

#### Proof of Theorem 2

**Proof.**

## 4. Proof of the Poincaré Inequalities for the Invariant Measure

**Proof.**

## 5. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Hodara, P.; Papageorgiou, I. Poincaré-Type Inequalities for Compact Degenerate Pure Jump Markov Processes. *Mathematics* **2019**, *7*, 518.
https://doi.org/10.3390/math7060518

**AMA Style**

Hodara P, Papageorgiou I. Poincaré-Type Inequalities for Compact Degenerate Pure Jump Markov Processes. *Mathematics*. 2019; 7(6):518.
https://doi.org/10.3390/math7060518

**Chicago/Turabian Style**

Hodara, Pierre, and Ioannis Papageorgiou. 2019. "Poincaré-Type Inequalities for Compact Degenerate Pure Jump Markov Processes" *Mathematics* 7, no. 6: 518.
https://doi.org/10.3390/math7060518