Let $M$ be a compact Riemann Surface. Let $P$ be a fixed point on $M$ and let $\delta_{P}$ be the Dirac point distribution at $M$. Consider the fundamental solution of the heat equation $$ (\partial_{t}-\Delta)K(x,y,t)=0, K(x,y,0)=\delta_{x}(y) $$ and its solution $$ f(x)=e^{\Delta}\delta_{P}(x)=\int K(x,y,1)\delta_{P}(y)dy=K(x,P,1) $$ I am wondering if there is any special meaning attached to the case when $t=1$ and $M$ is taken as a homogeneous space. The idea seems natural that one start Brownian motion with a pointed probability distribution and let time flows, then $f(x)$ can be used as a measure of the "distance" between $x$ and $P$. In the case $M=\mathbb{S}^{1}$ via its universal cover $\mathbb{R}^{1}$ we recovered a sum of Gaussians that resembles theta function.

However when I look up online, the explicit formulas we have are rather difficult to interpret. For $g\ge2$ they are quotients of $\mathbb{H}$, and thanks to Mckeans' work we know $$ K(x,y,t)=\frac{\sqrt{2}}{(4\pi t)^{3/2}}e^{-t/4 }\int^{\infty}_{p}\frac{se^{-s^2/4t}}{(\cosh(s)-\cosh(p))^{1/2}}ds,p=d_{\mathbb{H}}(x,y) $$ I do not really know any good interpretation of this when we fix $y=P,t=1$ and let $x$ vary and add up the copies corresponding to $\pi_{1}(M)$. In particular, I am curious what does this say related to the spectrum of the Laplacian in $M$. It should relate to analytic torsion of the Laplacian on the manifold, but so far I have not found any way to relate the two objects.