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I was wondering if there is any reference for the asymptotic expansion of heat kernel on a complete noncompact manifold. That is,

\begin{align} H(x,q,t) \sim \frac{e^{-\frac{d^2(q,x)}{4t}}}{(4\pi t)^{n/2}}\left(\sum_{j=0}^{\infty}t^j u_j(x,t)\right) \end{align} for any $x$ in a compact set and $t$ small. Here the expansion means \begin{align} H(x,q,t) - \frac{e^{-\frac{d^2(q,x)}{4t}}}{(4\pi t)^{n/2}}\left(\sum_{j=0}^{k}t^j u_j(x,t)\right)=O(t^{k+1-n/2}). \end{align}

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  • $\begingroup$ Do you assume your manifold to be of bounded geometry? If yes, I would say that the answer is yes, it comes out from how a heat kernel on a manifold can be constructed, and the keyword is "parametrix" (see a reference to Candel-Conlon in this answer — mathoverflow.net/questions/197892/… ) $\endgroup$ – Victor Kleptsyn Mar 16 '18 at 17:41
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The only reference I can offer you is the work by Vassilevich; Heat Kernel Expansion: User’s Manual

The chapter on Non-Integrable potentials may be of some use.

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