Decomposition of stochastic flows in manifolds with complementary distributions

TL;DR

This paper introduces a method to decompose stochastic flows on manifolds with complementary distributions into components acting along horizontal and vertical directions, enabling detailed analysis of their structure.

Contribution

It presents a novel decomposition of stochastic flows into horizontal and vertical diffeomorphism components, extending to a cascade decomposition in local coordinates.

Findings

Decomposition of stochastic flows into horizontal and vertical components.

Maximal cascade decomposition in local coordinates.

Component-wise analysis of stochastic flows.

Abstract

Let $M$ be a differentiable manifold endowed locally with two complementary distributions, say horizontal and vertical. We consider the two subgroups of (local) diffeomorphisms of $M$ generated by vector fields in each of of these distributions. Given a stochastic flow $\varphi_t$ of diffeomorphisms of $M$, in a neighbourhood of initial condition, up to a stopping time we decompose $\varphi_t = \xi_t \circ \psi_t$ where the first component is a diffusion in the group of horizontal diffeomorphisms and the second component is a process in the group of vertical diffeomorphisms. Further decomposition will include more than two components; it leads to a maximal cascade decomposition in local coordinates where each component acts only in the corresponding coordinate.