For a senior project of mine, I would like to know what the most general setting of Varadhan's formula for the geodesic distance in terms of the limiting behavior of heat kernels is. The result I'm talking about is

$\bf{Theorem.}$ $\it(Varadhan)$ Let $(M, g)$ be a member of some class of complete Riemannian manifolds. Let $K(t, x, y)$ be the associated heat kernel (converging to $\delta_x(y)$ at small time). Then \begin{equation} -4 t \text{ log } K(t, x, y) \to d(x,y)^2 \text{ as } t \to 0. \end{equation} where $d(x,y)^2$ is the squared distance function.

I skimmed through the original paper and the proof seems to only hold for manifolds with bounded distance functions. Clearly, however, with the heat kernel in $\mathbb{R}^n$ this result holds as well. On $\mathbb{R}^n$, the heat kernel can be given explicitly as

$K(t, x, y) = (4\pi t)^{-n/2} \text{exp}( {-\frac{1}{4t} |x - y|^2})$

and we see that

$-4 t \text{ log } K(t, x, y) \to |x - y|^2 \text{ as } t \to 0$

It is the motivating example for the theorem, after all! My knowledge of Riemannian manifolds and thereby heat kernels on these settings is only elementary (I am a senior undergrad), but I am under the impression that even considering heat kernels on manifolds that are not compact is an advanced topic. This makes me think there is a chance that there isn't a proper rephrasing of the theorem in a more general setting yet.

Regardless, I have tried a diligent search of the literature but cannot seem to find an explicit statement of a more general setting where the result holds. Some papers seem to proceed as if this result holds in a more general setting, yet I can't seem to find explicit statements of what setting it holds in. If you are able to give me a statement of a larger class of manifolds than the ones with bounded distance functions where this theorem holds, I would appreciate it! Also, references that I could cite would be ideal.

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    $\begingroup$ Interesting! Looking forward to answers. If you end up not getting an answer, though, you might consider asking this question on MathOverflow. $\endgroup$ – Jesse Madnick Apr 26 '15 at 23:10
  • $\begingroup$ Thank you for the tip! MathOverflow seems to be oriented towards professional mathematicians, so I was unsure if it would be appropriate for me to ask the question on that site or not. $\endgroup$ – user137769 Apr 26 '15 at 23:12
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    $\begingroup$ I located a set of lecture notes by Alexander Grigor'yan which mention this result in the general setting of connected manifolds. Maybe he is implicitly assuming the manifold is complete? But he puts a '~' instead of an '=' in front of the limit, so this still leaves me some room for doubt. The lecture notes are located here: math.uni-bielefeld.de/~grigor/cornell-lect.pdf. $\endgroup$ – user137769 Apr 27 '15 at 0:22
  • $\begingroup$ Hi, I'm trying to find the paper. Is there a pdf freely available somewhere? I can't manage to find it. $\endgroup$ – user8469759 Oct 17 '19 at 10:09
  • $\begingroup$ The paper by Varadhan I was referring to is “On the behavior of the fundamental solution of the heat equation with variable coefficients,” Communications on Pure and Applied Mathematics, vol. 20, no. 2, pp. 431–455, 1967. I believe I couldn't find it online either and had to use my university library system to obtain access. $\endgroup$ – user137769 Oct 18 '19 at 15:16

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