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I heard a talk where the speaker said that on a Riemannian manifold, for small values of $\text{dist }(x, y)$, the heat kernel $p_t(x, y)$ satisfies $$p_t(x, y) = \frac{1}{(4\pi t)^{n/2}}e^{-\frac{\text{dist }(x, y)^2}{4t}} + O(e^{-\frac{1}{\sqrt{t}}}).$$

Is this correct? Where can I find a reference for this fact? Thanks!

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I think a good reference is "Heat Kernels and Dirac Operators" by Nicole Berline, Ezra Getzler, Michèle Vergne, page 61.

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For some reason, I cannot locate it on page 61. Could you please tell me what edition you are using? Thanks for your help! – user85970 Jan 30 at 22:14
    
The one in Google Books: books.google.cz/… – Vít Tuček Jan 31 at 11:49
    
Okay, it was page 63 in the edition I could find. But I still don't understand why the expansion that we get for $p_t(x, y) - \frac{1}{(4\pi t)^{n/2}}e^{-\frac{\text{dist }(x, y)^2}{4t}} $ is actually $O(e^{-\frac{1}{\sqrt{t}}}).$ Is it obvious? I would really appreciate it if you could elaborate a bit. – user85970 Jan 31 at 12:47
    
@user85970 Sorry, I'm not an expert. You may find closely related discussion in this MO question: mathoverflow.net/questions/168438/… – Vít Tuček Jan 31 at 23:41

The original reference should be the paper by Molchanov "Diffusion processes and Riemannian geometry". You can find a pdf here (in russian):

http://www.mathnet.ru/links/74671d2aaeb444f56a570dddcce7e644/rm4123.pdf

and the english translation here:

http://iopscience.iop.org/article/10.1070/RM1975v030n01ABEH001400/meta

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