For standard Gaussian measure $\gamma$, the <Gaussian Poincaré inequality> is
$$
\operatorname{Var}_\gamma h\leq\int(h')^2d\gamma.
$$
The <Gaussian logarithmic Sobolev inequality> is
$$
\operatorname{Ent}_\gamma(h^2)
\leq2\int(h')^2d\gamma,
$$
where $\operatorname{Ent}_\gamma(G)=\int G\log G\,d\gamma-(\int Gd\gamma)\log\int Gd\gamma$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 208 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-208/3/a/solution