  In this paper we study a nonlocal Cahn-Hilliard model (CHE) in the framework
of random walk spaces, which includes as particular cases, the CHE on locally
finite weighted connected graphs, the CHE determined by finite Markov chains or
the Cahn-Hilliard Equations driven by convolution integrable kernels. We
consider different transitions for the phase and the chemical potential, and a
large class of potentials including obstacle ones. We prove existence and
uniqueness of solutions in $L^1$ of the Cahn-Hilliard Equation. We also show
that the Cahn-Hilliard equation is the gradient flow of the Ginzburg-Landau
free energy functional on an appropriate Hilbert space. We finally study the
asymptotic behaviour of the solutions.
