= Solution
Take $\lambda_0\to\infty$ with
$$
\boxed{D=\frac{s^2}{2\lambda_0},
\qquad \chi=\frac{sb}{\lambda_0}}
$$
fixed, so $s$ and $b$ scale as $\sqrt{\lambda_0}$. The rapidly relaxing flux satisfies $j\simeq-Dp_x+\chi c'p$, and therefore
$$
\boxed{p_t=\partial_x(Dp_x-\chi pc').}
$$
Zero stationary flux gives $p_s(x)\propto e^{\chi c(x)/D}$. A stationary probability density on $\mathbb R$ exists exactly when this exponential is integrable, in addition to the positivity condition $|c'|<\lambda_0/b$. The limiting process is
$$
\boxed{dX_t=\chi c'(X_t)dt+\sqrt{2D}\,dW_t.}
$$
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