Elliptic Feynman-Kac formula ($(L-V)u=-f$)

For a bounded domain, a bounded classical solution with boundary data $g$, nonnegative bounded $V$, bounded $f$ and integrable diffusion exit time $\tau$ satisfiesApply the Itô product rule to the discounted stopped solution and add its source integral. The stopped martingale is dominated by $\Vert u{\Vert }_{\infty }+\Vert f{\Vert }_{\infty }\tau$, so dominated convergence permits passage to the exit. For $V=f=0$, this is a boundary-value averaging principle for functions harmonic under $L$.