Brownian motion on Perelman's almost Ricci-flat manifold

TL;DR

This paper investigates Brownian motion and stochastic parallel transport on Perelman's almost Ricci-flat manifold, demonstrating convergence to Ricci flow objects and linking curvature bounds to Ricci flow characterizations.

Contribution

It constructs sequences of Brownian motions on Perelman's manifold that converge to Ricci flow counterparts and analyzes the limiting behavior via martingale problems.

Findings

Sequences of projected Brownian motions converge to Ricci flow objects.

Martingale problem analysis clarifies the limit process as N approaches infinity.

Curvature bounds lead to inequalities characterizing Ricci flow solutions.

Abstract

We study Brownian motion and stochastic parallel transport on Perelman's almost Ricci flat manifold $\mathscr M=M\times \mathbb S^N\times I$, whose dimension depends on a parameter $N$ unbounded from above. We construct sequences of projected Brownian motions and stochastic parallel transports which for $N \to \infty$ converge to the corresponding objects for the Ricci flow. In order to make precise this process of passing to the limit, we study the martingale problems for the Laplace operator on $\mathscr M$ and for the horizontal Laplacian on the orthonormal frame bundle $\mathscr{OM}$ . As an application, we see how the characterizations of two-sided bounds on the Ricci curvature established by A. Naber applied to Perelman's manifold lead to the inequalities that characterize solutions of the Ricci flow discovered by Naber and the second author.