Extension of time for decomposition of stochastic flows in spaces with complementary foliations

TL;DR

This paper extends the time interval during which stochastic flows on manifolds with complementary foliations can be decomposed into foliation-preserving diffeomorphisms, using techniques like Marcus equations and a 'stop and go' approach.

Contribution

It introduces methods to prolong the decomposition time of stochastic flows on manifolds with complementary foliations, including handling non-commuting vector fields and complex zones.

Findings

Extended the decomposition time for stochastic flows.

Applied Marcus equations for non-commuting vector fields.

Developed a 'stop and go' technique for general cases.

Abstract

Let $M$ be a manifold equipped (locally) with a pair of complementary foliations. In Catuogno, da 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 decomposed in diffeomorphisms that preserves this foliations. In this article we present techniques which allows us to extend the time of this decomposition. For this extension, we use two techniques: In the first one, assuming that the vector fields of the system commute with each other, we apply Marcus equation to jump nondecomposable diffeomorphisms. The second approach deals with the general case: we introduce a `stop and go' technique that allows us to construct a process that follows the original flow in the `good zones' for the decomposition, and remains paused in `bad zones'. Among other applications, our results open the possibility of studying the asymptotic behaviour of each component. \end{abstract}