Topology of Foliations and Decomposition of Stochastic Flows of Diffeomorphisms

Author(s)

Melo, Alison M. ^[1] ; Morgado, Leandro ^[1] ; Ruffino, Paulo R. ^[1]

Affiliation

^[1] Univ Estadual Campinas, Dept Matemat, BR-13083859 Campinas, SP - Brazil

• Document type: Journal article
• Source: Journal of Dynamics and Differential Equations; v. 30, n. 1, p. 39-54, MAR 2018.
• Web of Science Citations: 0

Abstract

Let M be a compact manifold equipped with a pair of complementary foliations, say horizontal and vertical. In Catuogno et al. (Stoch Dyn 13(4):1350009, 2013) it is shown that, up to a stopping time , a stochastic flow of local diffeomorphisms 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-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. (AU)

• FAPESP's process: 11/50151-0 - Dynamical phenomena in complex networks: fundamentals and applications
• Grantee: Elbert Einstein Nehrer Macau
• Support Opportunities: Research Projects - Thematic Grants
• FAPESP's process: 11/14797-2 - Stochastic dynamics in foliated spaces
• Grantee: Leandro Batista Morgado
• Support Opportunities: Scholarships in Brazil - Doctorate (Direct)
• FAPESP's process: 12/18780-0 - Geometry of control systems, dynamical and stochastics systems
• Grantee: Marco Antônio Teixeira
• Support Opportunities: Research Projects - Thematic Grants