= Elliptic Feynman-Kac formula
{title2=$(L-V)u=-f$}
For a bounded domain, a bounded <classical solution> with boundary data $g$, nonnegative bounded $V$, bounded $f$ and integrable diffusion exit time $\tau$ satisfies
$$
u(x)=\mathbb E_x\left[e^{-\int_0^\tau V(X_r)dr}g(X_\tau)+\int_0^\tau e^{-\int_0^sV(X_r)dr}f(X_s)ds\right].
$$
Apply the <Itô product rule> to the discounted stopped solution and add its source integral. The stopped martingale is dominated by $\|u\|_\infty+\|f\|_\infty\tau$, so <dominated convergence> permits passage to the exit. For $V=f=0$, this is a boundary-value averaging principle for functions harmonic under $L$.
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