  Let $M$ be a compact manifold equipped with a pair of complementary
foliations, say horizontal and vertical. In Catuogno, Silva and Ruffino
($Stoch$. $Dyn$., 2013) it is shown that, up to a stopping time $\tau$, a
stochastic flow of local diffeomorphisms $\varphi_t$ in $M$ can be written as a
Markovian process in the subgroup of diffeomorphisms which preserve the
horizontal foliation composed with a process in the subgroup of diffeomorphisms
which preserve the vertical foliation. Here, we discuss topological aspects of
this decomposition. The main result guarantees the global decomposition of a
flow if it preserves the orientation of a transversely orientable foliation. In
the last section, we present an It\^o-Liouville formula for subdeterminants of
linearised flows. We use this formula to obtain sufficient conditions for the
existence of the decomposition for all $t\geq 0$.
