1. Topology of foliations and decomposition of stochastic flows of diffeomorphisms
- Author
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Melo, Alison M., Morgado, Leandro, and Ruffino, Paulo R.
- Subjects
Mathematics - Dynamical Systems ,Mathematics - Probability ,60H10, 58J65, 57R30 - Abstract
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 $t\geq 0$.
- Published
- 2015