= Zero-noise limit with a bounded potential
{title2=$u^\varepsilon(t,x)\to f(x_t)\exp\!\left(\int_0^t c(x_s)ds\right)$}
For globally Lipschitz drift $b$, couple $dX^\varepsilon=b(X^\varepsilon)dt+\varepsilon dW$ and $\dot x=b(x)$ from the same point. The <Gronwall inequality> gives $\sup_{s\le t}|X_s^\varepsilon-x_s|\le\varepsilon e^{Lt}\sup_{s\le t}|W_s|$. Thus convergence is uniform on each finite horizon. The <Feynman-Kac formula> for generator $\varepsilon^2\Delta/2+b\cdot\nabla+c$ has weight $\exp\int c$, with a positive sign. Bounded continuous $c$ is uniformly continuous on a compact neighbourhood of the deterministic trajectory, and bounded continuous $f$ supplies a deterministic integrable bound. The <dominated convergence theorem> proves the displayed limit without requiring global uniform continuity of $c$.
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