Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 202 / 3

Apply Itô formula to $\varphi (x)=x\log x$. Since ${\varphi }^{\prime \prime }(x)=1/x$,Boundedness of $\log M$ makes the stochastic integral a true martingale of mean zero. Taking expectations proves

The heat-semigroup form is $U(t,x)=P_{1-t}(f^2)(x)$. For $s<t$, independence and additivity of Brownian increments givewhere ${\tilde{B}}_{1-s}$ is an independent $N(0,1-s)$ increment. Thus $M_t=U(t,W_t)$ is a martingale.

The backward heat equation and Itô formula giveMoreover $M_1=f(W_1{)}^2$ and $M_0=\mathbb{E}f(W_1{)}^2$. Substitution into part a yields

The supplied derivative identity and the Cauchy-Schwarz inequality giveThereforeThe sum $W_t+B_{1-t}$ is standard normal for every $t$, so the right side is $2\mathbb{E}{f}^{\prime }(W_1{)}^2$. Part c proves the Gaussian logarithmic Sobolev inequality