Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 353 / 2 / a / iii / Solution

Let $Z={\int }_X{e}^{-V}dx$ and $P_B=e^{-V}/Z$. ThenBy nonnegativity of Kullback-Leibler divergence,with equality exactly when $P=P_B$ almost everywhere. Hence a normalizable $P_B$ is the unique minimizer and, because $\dot{F}<0$ elsewhere, the unique steady density compatible with the boundary condition.