# Concentration and Poincar\'e type inequalities for a degenerate pure jump Markov process

## TL;DR

This paper investigates concentration and Poincaré inequalities for a class of degenerate pure jump Markov processes, providing exponential convergence rates relevant for modeling biological neural networks.

## Contribution

It introduces new concentration and Poincaré inequalities for degenerate jump processes with memory, extending existing results to more complex neural network models.

## Key findings

- Established exponential convergence to equilibrium
- Derived new inequalities for degenerate jump processes
- Applied results to biological neural network models

## Abstract

We study Talagrand concentration and Poincar\'e type inequalities for unbounded pure jump Markov processes. In particular we focus on processes with degenerate jumps that depend on the past of the whole system, based on the model introduced by Galves and L\"ocherbach in \cite{G-L}, in order to describe the activity of a biological neural network. As a result we obtain exponential rates of convergence to equilibrium.