4
$\begingroup$

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!

$\endgroup$

2 Answers 2

2
$\begingroup$

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

$\endgroup$
4
  • $\begingroup$ For some reason, I cannot locate it on page 61. Could you please tell me what edition you are using? Thanks for your help! $\endgroup$
    – user85970
    Jan 30, 2016 at 22:14
  • $\begingroup$ The one in Google Books: books.google.cz/… $\endgroup$ Jan 31, 2016 at 11:49
  • $\begingroup$ 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. $\endgroup$
    – user85970
    Jan 31, 2016 at 12:47
  • $\begingroup$ @user85970 Sorry, I'm not an expert. You may find closely related discussion in this MO question: mathoverflow.net/questions/168438/… $\endgroup$ Jan 31, 2016 at 23:41
2
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.