As far as I know (I am not an expert), any solution of the heat equation on $\mathbb{R}^n$ or on a closed Riemannian manifold with the given initial condition can be presented in terms of stochastic integral (Feynman-Kac formula).

I am wondering if there is a generalization of the Feynman-Kac formula to compact Riemannian manifolds with boundary.

More precisely let $M$ be a compact Riemannian manifold with boundary $\partial M$ (e.g. a bounded domain with smooth boundary in $\mathbb{R}^n$). Let $$u\colon M\times [0,T]\to \mathbb{R}$$ be a solution of the heat equation $u_t=-\Delta u$ with the boundary conditions $u(x,0)=\phi(x)$, and $u(z,t)$ is a given function of $z\in\partial M,t\in[0,T]$. How to present $u$ in terms of the boundary data using a path integral?

The case of bounded domains in $\mathbb{R}^n$ is already new to me.

EDIT: As commented by Mateusz Kwasnicki, in the case of $\mathbb{R}^n$ there are extra difficulties. As far as I understand, they are related to the growth condition of a solution at infinity. Thus at the moment I prefer to restrict the discussion to **compact** manifolds, e.g. bounded domains in $\mathbb{R}^n$ with smooth boundary.

everysolution has a probabilistic interpretation, just as noteverysolution is given in terms of the heat kernel. (2) Feynman–Kac originally referred to solutions of the heat (or Poisson) equation with a potential; formulae of the form $u(t,x) = \mathbb{E}^x u_0(W_t)$, where $W_t$ is the Wiener process (with absorption or reflection on the boundary, if any), are much older and, if I am not mistaken, attributed to Kakutani. $\endgroup$ – Mateusz Kwa?nicki Mar 17 at 0:22