Let $D \subseteq \mathbb{R}^m$ be a bounded domain. The Feynman-Kac formula for the heat equation with initial condition $u(t, x) = f(x)$ and boundary condition $u(t, x)|_{\partial D} = 0$ is given by
$$ u(t, x) = \mathbb{E}_x\left( f(B(t)) \chi(B)\right),$$ where $B(s), 0 \leq s \leq t$ is a Brownian path, and $\chi(B) = 1$ if $B([0, t]) \subseteq D$ and $0$ otherwise. In other words, $\chi$ ensures that the path has not hit the boundary $\partial D$ within time $t$, which corresponds to the zero Dirichlet boundary condition.
My question: is there a corresponding version of the Feynman-Kac where the boundary condition is not zero? For starters, let's take the easiest situation, $u(t, x)|_{\partial D} = c$, a non-zero constant.
I cannot find such a statement in the literature, nor can I come up with a formulation. However, when I think physically, I cannot also come up with a heuristic reasoning as to why such a formula cannot exist. Physically, heat is still being conducted across the boundary, only the boundary temperature is different. Is there a physical justification as to why there should not be a microscopic formulation for this situation, like "at absolute zero, particles stop vibrating totally", or something along those lines?
