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Consider Varadhan's famous formula for the kernel of the heat equation on a manifold:

$$ \lim_{t \rightarrow 0} t \log h(t,x,y) = - \frac{d(x,y)^2}{4} .$$

I do not have access to his 1967 two papers, but from what I can understand Varadhan obtained this as a byproduct of a much more general probabilistic result (something about stochastic processes, apparently) and by using probabilistic methods and concepts, hence my question: has anybody found in the meantime a more straightforward, purely geometric (Riemannian) proof of the above formula? (I would be happy even with a slightly weaker result than the above, but not much weaker.)

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I would say that yes. In fact, a nearby question is "why the heat kernel on a Riemannian manifold exists" (well, it depends on what do you prefer to construct first: the Brownian motion [say, out of discrete random walk], then using it to solve the heat equation, or the heat kernel, then using it to define the Brownian motion).

Both questions can be answered simultaneously via an explicit construction. You can find it in Candel, Conlon, Foliations II, 2003, page 441, and the keyword is "parametrix". It is an "almost-solution": an explicit function $H(t,x,y)$, that is singular at $t=0$, but sufficiently well approximates the heat kernel so that $\partial_t H - \Delta H$ is a nice function $K$. Then you can find the true heat kernel $h(t,x,y)$ by correcting $H$ by a series of convolutions (in both space and time): $H-H*(K+K*K-K*K*K+\dots)$. And out of this, you get everything.

Well, it's perhaps a bit too heavy artillery for such a simple question (normally, you were asking only for that a heat kernel on a manifold behaves in the same was as in $\mathbb{R}^n$, where it is given by an explicit formula). But at least it works.

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Varadhan's result was extended to wider class of operators and manifolds by Molchanov: http://iopscience.iop.org/0036-0279/30/1/R01 (Diffusion processes and Riemannian geometry). See also http://arxiv.org/abs/math/0601350 (Small time asymptotics of diffusion processes, by A.F.M. ter Elst, D.W. Robinson and A. Sikora) where other relevant references can be found.

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    $\begingroup$ Agreed, but these approaches too are essentially probabilistic, not purely geometric. $\endgroup$ – Alex M. Feb 20 '15 at 10:18
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Varadhan's paper is analytic, he writes in the acknowledgement that the first version of his paper was probabilistic, but that he was able to replace those arguments by analytic ones.

You can find his paper at http://sci-hub.se/https://doi.org/10.1002/cpa.3160200210

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