We study Brownian motion and stochastic parallel transport on Perelman’s almost Ricci flat manifold $ℳ=M\times {𝕊}^N\times I$, 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 $ℳ$ and for the horizontal Laplacian on the orthonormal frame bundle $𝒪ℳ$. 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.

中文翻译：

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Perelman几乎Ricci平流形上的布朗运动

我们研究Perelman几乎Ricci平面流形上的布朗运动和随机平行传输 $ℳ=\mathrm{中号}\times {𝕊}^{ñ}\times \mathrm{一世}$，其尺寸取决于从上方无界的参数N。我们构造了预测的布朗运动和随机平行传输的序列，它们对于$ñ\to \infty$收敛到Ricci流的相应对象。为了使传递过程达到极限的精确性，我们研究了Laplace算子的on问题。$ℳ$ 对于正交框架束上的水平拉普拉斯算子 $𝒪ℳ$。作为一个应用程序，我们看到A. Naber在Perelman流形上建立的Ricci曲率两侧边界的刻画如何导致表征Naber和第二作者发现的Ricci流解的不等式。