Equivariant diffusions on Principal bundles

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.