Ricci flow as a stochastic process

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.

4 Responses to Ricci flow as a stochastic process

1. hermann says:

Maybe this paper could be interesting:

Click to access 0809.0957v4.pdf

2. mfrasca says:

Indeed there is a number of people working in the application of Hamilton and Perelman ideas to physics. It is also clear that this beautiful mathematical theory has the charm of a deep truth and so we should see it somewhere in our quest for quantum gravity. As you read in this manuscript the author has to stop his understanding of a general Ricci flow as a stochastic process when he recognizes the appearing of singularities in the flow. This problem motivated all the work of Perelman and his proof of the geometrization conjecture. This is also the reason why I have put this as a conjecture. It is only my guess but I think that the proof of this conjecture will not be an easy task and will produce a lot of interesting understanding for quantum gravity.

Marco

3. Rafael says:

Dear Marco,

I came across this beautiful paper from Moffat that I guess sets the answer about what is the stochastic process (or quantum dynamics if you wish) below GR:
http://xxx.lanl.gov/abs/gr-qc/9610067
Regards,

Rafael.

4. mfrasca says:

Dear Rafael,

I was aware of Moffat’s works since long time. I think his approach should be seen in the same vein of stochastic electrodynamics where one tries to reproduce QED introducing stochastic terms into Maxwell equations. I have not had any infos about the success of these kind of approaches and their shortcomings and so I cannot add anything more to the discussion.

Regards,

Marco

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