# Long Time Behaviour on a Path Group of the Heat Semi-group Associated to a Bilaplacian

## Abstract

**:**

## 1. Introduction

**strictly positive**heat kernel

**change sign**. We have still when $t\to \infty $

**if and only if**the group is locally compact. Haar measure means that for all bounded measurable function $\tilde{F}$

**distributional**sense) ${Q}_{t}^{k}$ satisfies the three next properties:

- (1)
- ${Q}_{t}^{k}\Psi (\sigma )$ is still in the considered space
- (2)
- ${Q}_{t}^{k}\circ {Q}_{{t}^{\prime}}^{k}={Q}_{t+{t}^{\prime}}^{k}$
- (3)
- When $t\to \infty $$${Q}_{t}^{k}\Psi (\sigma )({g}_{.})\to {\int}_{C(]0,1],G)}\Psi (\sigma )dD$$

## 2. A Brief Review on the Haar Distribution on a Path Group

**Definition**

**1:**

**Theorem**

**2:**

**Theorem**

**3:**

**Theorem**

**4:**

**Theorem**

**5:**

## 3. A Non-Markovian Semi-group on a Path Group

**Remark:**

**Theorem**

**6:**

**Proof:**

**Definition**

**7:**

**lemma**

**8:**

**Proof:**

**Theorem**

**9:**

**Proof:**

## 4. Long Time Behaviour

**Theorem**

**10:**

**Proof:**

## 5. Conclusions

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Leandre, R.
Long Time Behaviour on a Path Group of the Heat Semi-group Associated to a Bilaplacian. *Symmetry* **2011**, *3*, 72-83.
https://doi.org/10.3390/sym3010072

**AMA Style**

Leandre R.
Long Time Behaviour on a Path Group of the Heat Semi-group Associated to a Bilaplacian. *Symmetry*. 2011; 3(1):72-83.
https://doi.org/10.3390/sym3010072

**Chicago/Turabian Style**

Leandre, Remi.
2011. "Long Time Behaviour on a Path Group of the Heat Semi-group Associated to a Bilaplacian" *Symmetry* 3, no. 1: 72-83.
https://doi.org/10.3390/sym3010072