# Nonlocal diffusion equations in Carnot Groups

## TL;DR

This paper investigates nonlocal diffusion equations on Carnot groups and demonstrates that, under suitable rescaling, solutions approximate local Dirichlet problems using Taylor series expansions specific to Carnot groups.

## Contribution

It introduces a method to approximate local PDE solutions on Carnot groups via nonlocal equations with rescaled kernels, utilizing Taylor series expansions in this geometric setting.

## Key findings

- Solutions converge uniformly to local PDE solutions as epsilon approaches zero.
- The approach employs Taylor series development tailored to Carnot group geometry.
- The method provides a bridge between nonlocal and local diffusion models in sub-Riemannian spaces.

## Abstract

Let $G$ be a Carnot group. We study nonlocal diffusion equations in a domain $\Omega$ of $G$ of the form $$ u_t^\epsilon(x,t)=\int_{G}\frac{1}{\epsilon^2}K_{\epsilon}(x,y)(u^\epsilon(y,t)-u^\epsilon(x,t))\,dy, \qquad x\in \Omega $$ with $u^\epsilon=g(x,t)$ for $x\notin\Omega$. For appropriate rescaled kernel $K_\epsilon$ we prove that solutions $u^\epsilon$, when $\epsilon\rightarrow0$, uniformly approximate the solution of different local Dirichlet problem in $G$. The key tool used is the Taylor series development for a function defined on a Carnot group.