Topology of foliations and decomposition of stochastic flows of diffeomorphisms

TL;DR

This paper explores the topological conditions under which stochastic flows of diffeomorphisms on manifolds can be globally decomposed according to foliations, extending previous local results and providing criteria for global decomposition.

Contribution

It establishes conditions for the global decomposition of stochastic flows preserving foliation orientation and introduces an Itô-Liouville formula for subdeterminants of linearized flows.

Findings

Global decomposition guaranteed if flow preserves foliation orientation.

Presented an Itô-Liouville formula for subdeterminants.

Provided sufficient conditions for decomposition for all time.

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 $t\geq 0$.