The heat kernel as an exponential of an integral In $\mathbb{R}^n$, if $\gamma$ is a line segment between $x_0 = \gamma (0)$ and $x = \gamma (t)$, one has the following formula:
$$\frac {\mathbb{e}^{- \frac{1}{4} \int_0^t <\dot{\gamma}, \dot{\gamma}> \mathbb{d}s}}{\sqrt {4 \pi t}^n} = h(t,x,x_0) ,$$
where $h$ is the kernel of the heat equation centered in $x_0$ at time $0$.
Now, if $(M,g)$ is a Riemann manifold (with the appropriate conditions for the heat kernel to exist), is there any curve $\gamma$ that could play a similar role to the line segment in the above formula (possibly with a different denominator)?
The only formula that I know of is
$$\mathbb{e}^{- \frac{1}{4} \int_0^t <\dot{\gamma}, \dot{\gamma}> \mathbb{d}s} = \mathbb{e}^ {- \frac{d(x,x_0)^2}{4 t}} ,$$
where $d$ is the distance associated to $g$ and $\gamma$ is a minimizing geodesic. This is nice, but quite far from the heat kernel.
 A: Actually, I think your second formula might indeed relate the distance on a Riemannian manifold to the heat kernel - see this paper by Varadhan, in which it is shown that if $p(t,x,y)$ is a solution to
$\frac{\partial p}{\partial t} = \frac{1}{2} \sum_{i,j=1}^k a_{ij}(x) \frac{\partial^2 p}{\partial x^i \partial x^j}$
such that $p(t,x,y) \to \delta_x(y)$ as $t \to 0$, then
$\lim_{t \to 0} -2t \log p(t,x,y) = d(x,y)^2,$
where $d$ is distance in the Riemannian metric derived from the coefficients $\left\{ a_{ij}(x) \right\}$.
(the link below is to a preview of the paper, but the relevant equation (what I gave above) is right on the front page):
http://www.readcube.com/articles/10.1002%2Fcpa.3160200210?r3_referer=wol&tracking_action=preview_click&show_checkout=1
I found this from another paper by Saloff-Coste (available in full here):
http://www.math.cornell.edu/~lsc/hk.pdf
which in turn I found by some strategic googling because it looked to me as though you might be able to get somewhere by viewing the inner product $\langle \dot{\gamma}, \dot{\gamma} \rangle$ in your exponent as a Dirichlet form (see e.g. Dirichlet Forms and Symmetric Markov Processes by Fukushima, Oshima, and Takeda).
Hope this helps!
