  A cyclic random walk is a random walk whose transition probabilities/rates
can be written as a superposition of the empirical measures of a family of
finite cycles. This identifies a convex set of models. We discuss the problem
of characterization of cyclic random walks in some special cases showing that
it is related to several remarkable and classical results. In particular we
introduce the notion of balanced measure and show that a translation invariant
random walk on $\mathbb Z^d$ is cyclic if and only if its transition
probability is balanced. The characterization of the extremal elements is
obtained using the Carath\'{e}odory's Theorem of convex analysis. We then show
that a random walk on a finite set is cyclic if and only if at every vertex the
outgoing flux of the transition graph is equal to the ingoing flux. The
extremal elements are characterized by the Birkhoff-Von-Neumann Theorem.
Finally we consider the discrete torus and discuss when the cyclic
decomposition can be done using only homotopically trivial cycles or elementary
cycles associated to edges and two dimensional faces. While in one dimension
this is equivalent to require some geometric properties of a discrete vector
field associated to the transition rates, in two dimension this is not the
case. In particular we give a simple characterization of the polyhedron of the
rates admitting a cyclic decomposition with elementary cycles. The proof is
based on a discrete Hodge decomposition, elementary homological algebra and the
Helly's Theorem of convex analysis. Finally we discuss a natural discretization
procedure of smooth divergence free continuous vector fields and an application
to random walks in random environments.
