Ricci flow as a stochastic process

30/01/2009

Yesterday I have posted a paper on arxiv (see here). In this work I prove a theorem about Ricci flow. The question I give an answer is the following. When you have a heat equation you have always a stochastic process from which such an equation can be derived. In two dimensions the Ricci flow takes the straightforward form of a heat equation. So, could it be derived from a stochastic process? The answer is affirmative and can be obtained through a generalization of path integrals (Wiener integrals) on a Riemannian manifold given here. One can write for the metric something like

g=\int [dq]\exp[-{\cal L}(q)]g_0

so, what is \cal L? The really interesting answer is that this is Perelman \cal L-length functional. A similar expression was derived by Bryce DeWitt in the context of Feynman’s path integrals in a non-Euclidean manifold in 1957 (see here) but in this case we are granted of the existence of the integral.

This result shows a really interesting conclusion that underlying Ricci flow there is a stochastic process (Wiener process), at least in two dimensions. So, we propose a more general conjecture: Ricci flow is generated by a Wiener process independently on the dimensionality of the manifold.

I’ll keep on working on this as this result provide a clear path to quantum gravity. Mostly, I would like to understand how Ricci flow and the non-linear sigma model are connected. Also here, I guess, Perelman will play a leading role.


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