Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 356 / 3 / c / Solution

Take ${\lambda }_0\to \infty$ withfixed, so $s$ and $b$ scale as $\sqrt{{\lambda }_0}$. The rapidly relaxing flux satisfies $j\simeq -D{p}_x+\chi c^{\prime }p$, and thereforeZero 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 $|c^{\prime }|<{\lambda }_0/b$. The limiting process is