Path integral solution to heat equation Let $(M,g)$ be a compact Riemannian manifold. Then the solution $u:M\times [0,\infty)\to \mathbb{R}$ of the heat equation $\partial_t u=\Delta_gu$ starting at $u_0\in C^{\infty}(M)$ is given by the path integral
$$
u(x,t)=\int_{\gamma \in H_x} e^{-E(\gamma)/4t}u_0(\gamma(1))d\gamma,
$$
where the integral is taken over the classical Wiener space $H_x\subset L^{2,1}([0,1],M)$ of finite energy paths $\gamma:[0,1]\to M$ starting at $x$ (i.e. $\gamma(0)=x$). Also, here 
$$
E(\gamma)=\int_0^1\Big|\frac{d\gamma}{ds}\Big|^2ds
$$ 
is the Dirichlet energy of a curve in $(M,g)$ and the measure of integration is the Wiener measure. See this article for a survey of the above.
Question: Does the above path integral formula for $u$ hold for more general parabolic PDEs?
More precisely, let $L:C^{\infty}(M)\to C^{\infty}(M)$ be a second order elliptic operator (e.g. above we took $L=\Delta_g$).
Then can we write the solution $u:M\times [0,T)\to \mathbb{R}$ of the parabolic PDE 
$$
\partial_tu=Lu
$$
with initial condition $u_0\in C^{\infty}(M)$ as the path integral
$$
u(x,t)=\int_{\gamma \in H_x} e^{-S(\gamma)/4t}u_0(\gamma(1))D\gamma,
$$
for some functional $S:H_x\to \mathbb{R}$ on the path space? What is $S$ in this case? Note that $S=E$ when $L=\Delta$.
 A: I am not sure if this is your point, but the path integral you mentioned is, at least to me, as if it refers to stochastic differential equations (SDE) and the Fokker-Planck equation.
To help explain the mechanism behind the Fokker-Planck equation, let us start from a naïve example. For the heat equation in 1-D Euclidean space $\mathbb{R}$, consider the following Brownian motion
$$
{\rm d}X_t=\sigma\,{\rm d}W_t,
$$
where the constant $\sigma$ denotes the thermal diffusivity, the stochastic process $W_t$ means the Wiener process. Here $X_t$ is also a stochastic process, tracing the position of a Brownian particle at time $t$. Further, denote by $u(t,x)$ the probability density function (PDF) of $X_t$, i.e.,
$$
\int_{-\infty}^xu(t,y)\,{\rm d}y=\mathbb{P}(X_t\le x).
$$
We hope to find a differential equation that governs $u=u(t,x)$.
Define
$$
f(x)=e^{-2\pi i\xi x}.
$$
Then
$$
\mathbb{E}f(X_t)=\int_{\mathbb{R}}e^{-2\pi i\xi x}u(t,x)\,{\rm d}x=\hat{u}(t,\xi)
$$
is the Fourier transform of the PDF. This is known as the characteristic function in probability. To find a governing equation for $u$, it suffices to find that for $\hat{u}$.
By Ito's formula, we have
\begin{align}
{\rm d}f(X_t)&=f'(X_t)\,{\rm d}X_t+\frac{1}{2}f''(X_t)\,{\rm d}\left<X\right>_t\\
&=-2\pi^2\xi^2\sigma^2f(X_t)\,{\rm d}t-2\pi i\xi\sigma f(X_t)\,{\rm d}W_t,
\end{align}
where $\left<X\right>_t$ denotes the quadratic variation process of $X_t$, which, for the SDE stated above, reads
$$
{\rm d}\left<X\right>_t=\sigma^2\,{\rm d}t.
$$
From this result, we obtain
$$
f(X_t)=f(X_0)-2\pi^2\xi^2\sigma^2\int_0^tf(X_s)\,{\rm d}s-2\pi i\xi\sigma\int_0^tf(X_s)\,{\rm d}W_s.
$$
Note that the last integral is a martingle (see here as well), for which
$$
\mathbb{E}\left(\int_0^tf(X_s)\,{\rm d}W_s\right)=0.
$$
Thanks to this result, taking the expectation on both sides yields
\begin{align}
\mathbb{E}f(X_t)&=\mathbb{E}f(X_0)-2\pi^2\xi^2\sigma^2\mathbb{E}\left(\int_0^tf(X_s)\,{\rm d}s\right)\\
&=\mathbb{E}f(X_0)-2\pi^2\xi^2\sigma^2\int_0^t\mathbb{E}f(X_s)\,{\rm d}s,
\end{align}
or using $\mathbb{E}f(X_t)=\hat{u}(t,\xi)$,
$$
\hat{u}(t,\xi)=\hat{u}(0,\xi)-2\pi^2\xi^2\sigma^2\int_0^t\hat{u}(s,\xi)\,{\rm d}s,
$$
which is equivalent to the differential form
$$
\frac{\rm d}{{\rm d}t}\hat{u}(t,\xi)=-2\pi^2\xi^2\sigma^2\hat{u}(t,\xi),
$$
whose inverse Fourier transform gives
$$
\frac{\partial}{\partial t}u(t,x)=\frac{\sigma^2}{2}\frac{\partial^2}{\partial x^2}u(t,x).
$$
To sum up, the solution of a heat equation can be interpreted as the probability density of some stochastic process.
This concept can be generalized to the general Ito process, a special type of stochastic processes governed by
$$
{\rm d}X_t=\mu(t,X_t)\,{\rm d}t+\sigma(t,X_t)\,{\rm d}W_t,
$$
where $\mu$ and $\sigma$ are both preloaded functions. Repeat the above derivation with further techniques such as
$$
\mathbb{E}\left(f(X_t)\,\mu(t,X_t)\right)=\int_{\mathbb{R}}e^{-2\pi i\xi x}\mu(t,x)u(t,x)\,{\rm d}x=\widehat{\left(\mu\,u\right)}(t,\xi),
$$
one may end up with the general 1-D Fokker-Planck equation
$$
\frac{\partial}{\partial t}u(t,x)=-\frac{\partial}{\partial x}\left(\mu(t,x)\,u(t,x)\right)+\frac{1}{2}\,\frac{\partial^2}{\partial x^2}\left(\sigma^2(t,x)\,u(t,x)\right).
$$
One may also generalize this 1-D equation to $n$-D cases by employing an $n$-dimensional SDE system, as is stated in this page. As far as I see, one may also generalize this Euclidean-space result to differentiable manifolds, but I am afraid it goes beyond my knowledge.
Hope this explanation could be somewhat helpful for your.
