Topology of foliations and decomposition of stochastic flows of diffeomorphisms

2015/11/04 by Alison M. Melo, Melo, Alison M., Leandro Morgado +3
Mathematics · #57R30 #58J65 #60H10 #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR) #math.DS #math.PR #msc:57R30 #msc:58J65 #msc:60H10

paper · pdf · doi:10.48550/arxiv.1511.01376

arxiv created 2015/11/04 · arxiv updated 2015/11/05

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 τ, a stochastic flow of local diffeomorphisms φ_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 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.

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