Stochastic interpretation of Einstein equations

The link between Einstein equations and a stochastic process can be achieved through Ricci flow. I can state the question very easily for the 2d case while, for higher dimensions things may become quite involved. The idea is that a stochastic process satisfy a diffusion equation

$$\partial_tP=\Delta_2P$$

and one can write down the solution through a Wiener integral

$$P=\int[dx(t)]e^{-\frac{1}{2}\int_0^td\tau{\dot x}^2(\tau)}.$$

When one extend this to a generic two-dimensional manifold, the diffusion equation, when applied to the metric, is that of the Ricci flow as one has just the Laplacian replaced by the Beltrami operator applied to metric. Then, the fixed point of this Ricci flow is just Einstein equations for the two-dimensional manifold at hand. I have given some considerations about, well-founded on a theorem by Baer and Pfaeffle (see here).

The exciting idea behind this is that a Ricci flow could be always derived from a stochastic process underlying a manifold. I think that this is material to be studied yet.