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Use one $d$-dimensional Brownian motion $W$ to couple the diffusion processesThe global existence theorem for stochastic differential equations with Lipschitz coefficients supplies a unique nonexplosive strong stochastic solution. Let $x$ be the deterministic drift trajectory. Subtraction and the Gronwall inequality give, on every finite horizon $t$,This also follows directly from the integral equations with continuous forcing, so the coupling can be chosen simultaneously for all $\epsilon$.

For completeness derive the needed Feynman-Kac formula with its potential sign. For fixed $t$, apply the Itô formula and the Itô product rule toIts drift is the exponential factor times $-u_t^{\epsilon }+({\epsilon }^2\Delta /2+b\cdot \nabla )u^{\epsilon }+c{u}^{\epsilon }$, which vanishes by the partial differential equation. It is a local martingale. Boundedness of $u^{\epsilon }$ on the finite time slab and boundedness of $c$ make $Y$ bounded there, so it is a true martingale. Its endpoint expectations giveThe deterministic trajectory on $[0,t]$ is compact. Uniform pathwise convergence puts all sufficiently small-noise paths in a fixed compact neighbourhood of that trajectory. Continuity of $c$ therefore implies ${\sup }_{s\le t}|c(X_s^{\epsilon })-c(x_s)|\to 0$; global uniform continuity of $c$ is not required. Continuity of $f$ gives convergence of the terminal factor. Finally,The dominated convergence theorem proves the zero-noise limit with a bounded potential:At $t=0$ this is the given initial condition. All boundedness arguments concern a fixed finite horizon; no uniform bound over infinite time is needed.