Stochastic perturbation of geodesics on the manifold of Riemannian metrics

TL;DR

This paper develops a diffusion process framework to analyze the evolution of Riemannian metrics driven by stochastic kinetic energy, providing new insights into geometric stochastic analysis.

Contribution

It introduces a novel diffusion-based approach to study the stochastic evolution of Riemannian metrics on the manifold.

Findings

Derived the evolution equation for Riemannian metrics under stochastic perturbations

Connected stochastic kinetic energy to geometric evolution equations

Provided a new method for analyzing stochastic processes on geometric manifolds

Abstract

In this paper, using diffusion processes, we compute the evolution equation on the manifold of Riemannian metrics for the Lagrangian induced by the $L^2$ stochastic kinetic energy functional.