Martingales on manifolds with time-dependent connection

TL;DR

This paper extends the theory of martingales to manifolds with time-dependent connections, exploring how properties change when the geometric structure varies over time.

Contribution

It introduces a framework for martingales on manifolds with evolving connections, broadening the scope of stochastic analysis in dynamic geometric settings.

Findings

Some properties of martingales extend to time-dependent connections

Not all fixed-connection properties hold in the time-dependent case

Provides foundational definitions for stochastic processes on evolving manifolds

Abstract

We define martingales on manifolds with time-dependent connection, extending in this way the theory of stochastic processes on manifolds with time-changing geometry initiated by Arnaudon, Coulibaly and Thalmaier (2008). We show that some, but not all properties of martingales on manifolds with a fixed connection extend to this more general setting.