Heat kernel asymptotic expansion on complete noncompact manifold

Question

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 \frac1{(4\pi t)^{n/2}}e^{-\frac{d^2(q,x)}{4t}}\sum_{j=0}^{\infty}t^j u_j(x,t) \end{align} for any $x$ in a compact set and $t$ small. Here the expansion means \begin{align} H(x,q,t) - \frac1{(4\pi t)^{n/2}}e^{-\frac{d^2(q,x)}{4t}}\sum_{j=0}^kt^j u_j(x,t)=O(t^{k+1-n/2}). \end{align}

Answer 1

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.

Answer 2

The answer is YES:

• Kannai, Y. Off diagonal short time asymptotics for fundamental solutions of diffusion equations. Comm. Partial Differential Equations 2 (1977), no. 8, 781–830, https://doi.org/10.1080/03605307708820048