= 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 $u(x)=\mathbb E_xu(X_{\tau_D})$. 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
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