 Bounded-domain harmonic uniqueness for a diffusion

For a bounded-domain diffusion with almost surely finite exit time ${\tau }_D$, bounded $L$-harmonic functions continuous up to the boundary satisfy $u(x)={\mathbb{E}}_{x}u(X_{{\tau }_D})$. Stop the test-function martingale and use bounded convergence. Identical boundary values therefore imply identical harmonic functions. Uniform ellipticity and bounded SDE coefficients ensure finite exit by the projection argument.

// Target: probability-and-statistics.bigb