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.
