# $(BV,L^p)$-decomposition, $p=1,2$, of Functions in Metric Random Walk Spaces

## TL;DR

This paper investigates the decomposition of functions into BV and L^p components within metric random walk spaces, encompassing applications like weighted graphs and image processing, and derives related variational equations and geometric insights.

## Contribution

It introduces the $(BV,L^p)$-decomposition framework for metric random walk spaces, deriving Euler-Lagrange equations, gradient flows, and geometric properties for $p=1,2$, including thresholding analysis.

## Key findings

- Derived Euler-Lagrange equations for the variational problems.
- Established gradient flow dynamics in the metric random walk setting.
- Analyzed geometric aspects and thresholding parameters for $p=1$.

## Abstract

In this paper we study the $(BV,L^p)$-decomposition, $p=1,2$, of functions in metric random walk spaces, a general workspace that includes weighted graphs and nonlocal models used in image processing. We obtain the Euler-Lagrange equations of the corresponding variational problems and their gradient flows. In the case $p=1$ we also study the associated geometric problem and the thresholding parameters.