  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.
