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}