This is a similar question to Heat Kernel Asymptotics on Manifold with Boundary. But I have some further questions.
Let $(M,g)$ be a closed Riemannian manifold, the heat kernel $p(x,y,t)$ of Laplace-Beltrami operator has the following local expansion $$p(x,y,t)\sim\frac{1}{(4 \pi t)^{n/2}}e^{d^2(x,y)/4t}\left(\sum_{j=0}^\infty t^ju_j(x,y)\right) $$ Where $u_0(x,y)=D^{-1/2}$, where $D$ is the volume density at $y$ reading in normal coordinates at $x$. And $u_j$ are recursively defined for $j\ge1$.
Now I would would like to ask about the similar expansion for manifolds with boundary and manifolds with conical singularities.
1 It was discussed in the aforementioned post that on a compact set $K\subseteq M$ that intersect boundary and the boundary is totally geodesic, there is an "0-th order" expansion when imposing the Dirichlet boundary condition.(The reference of this fact is not in that post, it will be more than helpful if someone could kindly point out a reference for this) $$\left|p(x,y,t)-\left(\frac{1}{ t^{n/2}}e^{-d^2(x,y)/4t}-\frac{1}{ t^{n/2}}e^{-\sigma^2(x,y)/4t}\right)u(x,y)\right|<\frac{C}{t^{n/2-\varepsilon}}$$ where $\sigma$ is the length of the shortest path from $x$ to $y$ that touches the boundary. (In the case where the the manifold has a double, y has a corresponding point $y^*$ then can one conclude that $\sigma(x,y)=d(x,y^*)$ for $d$ the distance of the double?). What does the geometric information encoded in $u$ given this expansion? Is it the same as in the case without boundary? What is the next term in the expansion, and what is its geometric meaning? Is there a possible $\sqrt{t}$ power in the next term?
Edit: the reason why I ask about the $\sqrt{t}$ power is that this survey https://arxiv.org/abs/hep-th/0306138 says on page 6 in the footnote that the heat kernel expansion itself can have a $\sqrt{t}$ power, this is not from integration, not like the case of heat trace expansion. However, this cannot happen in the interior, so I guess if it happens, it must happen on the boundary, but I couldn't even make an example.
2 Similarly if $M=[0,1]\times N$ with warped product metric $g_M=dr^2+r^2g_N$, i.e. $M$ is a finite cone with cross section $N$, then what does the heat kernel local expansion look like for a compact set containing cone tip? How does it related to the heat kernel on $N$?
