Topology of Foliations and Decomposition of Stochastic Flows of Diffeomorphisms.

Let <italic>M</italic> be a compact manifold equipped with a pair of complementary foliations, say horizontal and vertical. In Catuogno et al. (Stoch Dyn 13(4):1350009, <xref>2013</xref>) it is shown that, up to a stopping time τ<inline-graphic></inline-graphic>, a stochastic flow of local diffeomorphisms φt<inline-graphic></inline-graphic> in <italic>M</italic> 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 Itô-Liouville formula for subdeterminants of linearised flows. We use this formula to obtain sufficient conditions for the existence of the decomposition for all t≥0<inline-graphic></inline-graphic>. [ABSTRACT FROM AUTHOR]