Vertical martingales, stochastic calculus and harmonic sections

TL;DR

This paper introduces vertical martingales for smooth submersions, develops their stochastic calculus, and provides a stochastic characterization of harmonic sections, advancing the understanding of stochastic processes in differential geometry.

Contribution

It defines and constructs vertical martingales for smooth submersions and develops their stochastic calculus, offering new tools for studying harmonic sections.

Findings

Vertical martingales are constructed for smooth submersions.

A stochastic calculus for vertical martingales is developed.

Harmonic sections are characterized stochastically.

Abstract

This work is about a new class of martingales: the vertical martingales. We construct the vertical martingale for smooth submersions and we develop a stochastic calculus for one. Furthermore, we gives a stochastic characterization for harmonic sections.