  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.
