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

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.