# Modified log-Sobolev inequality for a compact PJMP with degenerate jumps

## TL;DR

This paper establishes a modified log-Sobolev inequality for a class of compact pure jump Markov processes modeling neuron interactions, leading to concentration results for empirical approximations.

## Contribution

It introduces a modified log-Sobolev inequality for a finite, compact PJMP with degenerate jumps, extending previous models of neuron interactions.

## Key findings

- Proves a modified log-Sobolev inequality for the process
- Derives concentration inequalities for empirical approximations
- Extends the model to include degenerate jumps in neuron interactions

## Abstract

We study the modified log-Sobolev inequality for a class of pure jump Markov processes that describe the interactions between brain neurons. In particular, we focus on a finite and compact process with degenerate jumps inspired by the model introduced by Galves and L\"ocherbach. As a result, we obtain concentration properties for empirical approximations of the process.