# The Total Variation Flow in Metric Random Walk Spaces

## TL;DR

This paper extends the Total Variation Flow to metric random walk spaces, unifying various graph and nonlocal models, and investigates their solutions, asymptotic behavior, geometric properties, and eigenvalue problems.

## Contribution

It introduces a general framework for TVF in metric random walk spaces, establishing existence, uniqueness, and analyzing geometric and spectral properties.

## Key findings

- Solutions reach the average of initial data in finite time for finite graphs.
- Introduces perimeter, mean curvature, and studies isoperimetric and Sobolev inequalities.
- Characterizes Cheeger and calibrable sets, and provides methods for the Cheeger cut problem.

## Abstract

In this paper we study the Total Variation Flow (TVF) in metric random walk spaces, which unifies into a broad framework the TVF on locally finite weighted connected graphs, the TVF determined by finite Markov chains and some nonlocal evolution problems. Once the existence and uniqueness of solutions of the TVF has been proved, we study the asymptotic behaviour of those solutions and, with that aim in view, we establish some inequalities of Poincar\'{e} type. In particular, for finite weighted connected graphs, we show that the solutions reach the average of the initial data in finite time. Furthermore, we introduce the concepts of perimeter and mean curvature for subsets of a metric random walk space and we study the relation between isoperimetric inequalities and Sobolev inequalities. Moreover, we introduce the concepts of Cheeger and calibrable sets in metric random walk spaces and characterize calibrability by using the $1$-Laplacian operator. Finally, we study the eigenvalue problem whereby we give a method to solve the optimal Cheeger cut problem.