# Asymptotic behaviour of fast diffusions on graphs

## TL;DR

This paper studies the long-term behavior of fast diffusions on finite directed graphs, analyzing how the diffusion process evolves as the speed increases and the likelihood of particles passing vertices decreases.

## Contribution

It provides new results on the asymptotic behavior of diffusion semigroups on graphs, extending previous duality approaches to $L^1$ and $L^2$ spaces.

## Key findings

- Asymptotic behavior characterized for increasing diffusion speeds.
- Diffusion semigroup convergence in $L^1$ and $L^2$ spaces.
- Results extend duality methods to new settings.

## Abstract

We investigate fast diffusions on finite directed graphs. We prove results in a way dual to presented in Bobrowski, A. Ann. Henri Poincar\'e (2012) 13(6): 1501-1510 and Bobrowski, A., Morawska, K. DCDS-B (2012), 17(7): 2313-2327, and obtain asymptotic behaviour of a diffusion semigroup on a graph in $ L^1 $ and $ L^2 $ as the diffusions' speed increases and the probability of a particle passing through a vertex decreases.