Equivariant diffusions on Principal bundles

TL;DR

This paper develops a framework for decomposing diffusion operators on principal bundles into horizontal and vertical components, enabling a better understanding of stochastic flows and their laws.

Contribution

It introduces a semi-connection on principal bundles for splitting diffusion operators, providing a new disintegration theorem and application to stochastic flow decomposition.

Findings

Decomposition of diffusion operators into horizontal and vertical parts.

Disintegration theorem for the law of the diffusion process.

Application to stochastic flow decomposition.

Abstract

Given a pair of second order diffusion operators, one on the total space of a principle bundle $N$ and the other on the base space $M$, intertwined by the projection $\pi:N\to M$, if the operator ${\mathcal A}$ on the base manifold has constant rank, we define a semi-connection on the principal bundle which allows to split the diffusion operator ${\mathcal B}$ on the total space into the sum of the horizontal lift of ${\mathcal A}$ and the other vertical. This allow to conclude a disintegration theorem for the law of ${\mathcal B}$. As an application, a decomposition of stochastic flow is given.