Ytukyg

Let $(M,g)$ be a compact Riemannian manifold. Then the solution $u:Mtimes [0,infty)to mathbb{R}$ of the heat equation $partial_t u=Delta_gu$ starting at $u_0in C^{infty}(M)$ is given by the path integral
$$
u(x,t)=int_{gamma in H_x} e^{-E(gamma)/4t}u_0(gamma(1))dgamma,
$$
where the integral is taken over the classical Wiener space $H_xsubset L^{2,1}([0,1],M)$ of finite energy paths $gamma:[0,1]to M$ starting at $x$ (i.e. $gamma(0)=x$). Also, here
$$
E(gamma)=int_0^1Big|frac{dgamma}{ds}Big|^2ds
$$
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 $u:Mtimes [0,T)to mathbb{R}$ of the parabolic PDE
$$
partial_tu=Lu
$$
with initial condition $u_0in C^{infty}(M)$ as the path integral
$$
u(x,t)=int_{gamma in H_x} e^{-S(gamma)/4t}u_0(gamma(1))Dgamma,
$$
for some functional $S:H_xto mathbb{R}$ on the path space? What is $S$ in this case? Note that $S=E$ when $L=Delta$.

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.,
$$
int_{-infty}^xu(t,y),{rm d}y=mathbb{P}(X_tle x).
$$
We hope to find a differential equation that governs $u=u(t,x)$.

Define
$$
f(x)=e^{-2pi ixi x}.
$$
Then
$$
mathbb{E}f(X_t)=int_{mathbb{R}}e^{-2pi ixi x}u(t,x),{rm d}x=hat{u}(t,xi)
$$
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<Xright>_t\
&=-2pi^2xi^2sigma^2f(X_t),{rm d}t-2pi ixisigma f(X_t),{rm d}W_t,
end{align}
where $left<Xright>_t$ denotes the quadratic variation process of $X_t$, which, for the SDE stated above, reads
$$
{rm d}left<Xright>_t=sigma^2,{rm d}t.
$$
From this result, we obtain
$$
f(X_t)=f(X_0)-2pi^2xi^2sigma^2int_0^tf(X_s),{rm d}s-2pi ixisigmaint_0^tf(X_s),{rm d}W_s.
$$
Note that the last integral is a martingle (see here as well), for which
$$
mathbb{E}left(int_0^tf(X_s),{rm d}W_sright)=0.
$$
Thanks to this result, taking the expectation on both sides yields
begin{align}
mathbb{E}f(X_t)&=mathbb{E}f(X_0)-2pi^2xi^2sigma^2mathbb{E}left(int_0^tf(X_s),{rm d}sright)\
&=mathbb{E}f(X_0)-2pi^2xi^2sigma^2int_0^tmathbb{E}f(X_s),{rm d}s,
end{align}
or using $mathbb{E}f(X_t)=hat{u}(t,xi)$,
$$
hat{u}(t,xi)=hat{u}(0,xi)-2pi^2xi^2sigma^2int_0^that{u}(s,xi),{rm d}s,
$$
which is equivalent to the differential form
$$
frac{rm d}{{rm d}t}hat{u}(t,xi)=-2pi^2xi^2sigma^2hat{u}(t,xi),
$$
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
$$
mathbb{E}left(f(X_t),mu(t,X_t)right)=int_{mathbb{R}}e^{-2pi ixi x}mu(t,x)u(t,x),{rm d}x=widehat{left(mu,uright)}(t,xi),
$$
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.