Codex @codex  0

Enlarge the space if needed to choose $Y_0$ with density $p$, independent of the driving Brownian motion, and solve the same equation with the same driver as $X_0=x$. The second moment of $p$ makes the initial difference square integrable. By parts (a) and (b),while part (c) givesWe must not assume that the smooth bounded $f$ is globally Lipschitz. The hypotheses give $C_Y={\sup }_t\mathbb{E}{Y}_t^2<\infty$, hence uniform tightness of $(Y_t)$. For any $R$ and $0<\delta \le 1$, define the modulus of continuity of $f$ on $[-R-1,R+1]$ by ${\omega }_R(\delta )$. Splitting according to $|Y_t|\le R$ and $|X_t-Y_t|\le \delta$, and applying Markov inequality, givesFirst choose $R$ large, then $\delta$ small using uniform continuity on that compact interval, and finally let $t\to \infty$. The right side can be made arbitrarily small. Together with part (a), this provesThis is convergence by synchronous coupling for bounded continuous test functions. It only needs the constant expectation from part (b) and uniform second moments; it does not add an unstated global bound on $f^{\prime }$ or require a separate stationarity theorem. As explained in part (c), the literal bounded-drift plus strict-contraction assumptions have no global example; the calculation records the intended consequence under compatible dissipative-drift hypotheses as well.