# Feynman-Kac formula for domains with boundary

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.

• 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. – Mateusz Kwa?nicki Mar 17 at 0:22
• 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. – Mateusz Kwa?nicki Mar 17 at 0:25
• @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? – MKO Mar 17 at 0:25
• I was thinking about Tychonoff's counterexample, an exotic solution in full space; see also Chung–Kim for a related construction. – Mateusz Kwa?nicki Mar 17 at 7:35