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.

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    $\begingroup$ I am not an expert either, but here are some thoughts: (1) Not every solution has a probabilistic interpretation, just as not every solution 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
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    $\begingroup$ The Wikipedia entry on stochastic processes and boundary value problems apparently confirms this. It is a pity, though, that it does not even link to the entry on the Feynman–Kac formula. In fact, both entries require attention of an expert. $\endgroup$ – Mateusz Kwa?nicki Mar 17 at 0:25
  • $\begingroup$ @MateuszKwa?nicki: Thank you. In your remark 1) do you refer to domains with boundary or it is also related to $\mathbb{R}^n$ and/or closed manfolds? $\endgroup$ – MKO Mar 17 at 0:25
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    $\begingroup$ I was thinking about Tychonoff's counterexample, an exotic solution in full space; see also Chung–Kim for a related construction. $\endgroup$ – Mateusz Kwa?nicki Mar 17 at 7:35

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